But they may not want to rush the moment. This collection of talent might not stick together much longer.
Deng has been a Windy City fixture for nearly a decade and arguably Rose's most productive teammate. A two-way contributor who's embraced a massive workload (39.1 minutes per game over the last three seasons), he's not the type of player teams would typically risk losing.
But his name has been a fixture on the trade rumor mill for several seasons, as recently as June's draft via ESPN's Marc Stein, and a number of potential free agents could offer Chicago an upgrade in talent.
Some of these players will become outright free agents at season's end, but others will have to exercise an early termination option in their current contract. Still another group will be headed for restricted free agency, meaning even mutual interest between those players and the Bulls might not be enough to get the job done.
The Bulls know the pitfalls of aiming high in free agency. They missed out on Tim Duncan, Tracy McGrady and Grant Hill in the summer of 2000, then came up empty in their pursuits of LeBron James, Dwyane Wade and Chris Bosh a decade later.
Without the proper expectation level in place, that same disappointment could plague Bulls fans next summer. Staying ahead of that curve here, I'll just say don't look for any LeBron James, Paul George, Dirk Nowitzki or Kobe Bryant sightings in Chicago next summer.
But there's still superstar talent beyond that fantastic four, all of whom would blend seamlessly alongside Rose.
For Carmelo Anthony to even become available to the Bulls, two things need to happen.
First, he'll need to walk away from the $23.2 million he's owed for the 2014-15 season. Still just 29 years old, though, he has at least one major payday left in his future, so those $23 million could be replaced and then some.
The second part of this equation isn't quite so simple. To get to Chicago, Anthony would have to leave the New York Knicks, something he said he has no intention of doing during an interview on Bloomberg Television's Market Markers, via Newsday's Al Iannazzone.
Outside of some inner duty to restore the Empire State to basketball relevance, though, what's going to keep Anthony in New York? An underwhelming offseason leaves the Knicks facing an uphill climb just to get back to the Eastern Conference semifinals. Anything beyond that is a long shot at best.
If the Bulls finally decide to amnesty Carlos Boozer next summer, Anthony could keep punishing defenders on the low block. Mike Dunleavy (career 37.2 percent three-point shooter) could be thrust into a starting gig to keep defenses honest, while rookie Tony Snell could be groomed to take over that spot down the line.
With a second scorer like Anthony, Rose's point totals might take a slight dip (career 21.0 per game). But his field-goal percentage would look closer to what he shot in his first two seasons (48.3) than what he did in his last two (44.2).
A Rose-Anthony duo remains highly unlikely, but it's a potentially brilliant combo if it comes to fruition.
Hopefully for his sake, that will be the end to his productive, underappreciated tenure there.
Once seen as a vital piece on a pair of title teams, his reputation has been battered over the last two seasons. An undeserving scapegoat, he's become the unfortunate face of everything wrong with that organization.
He's not a stretch forward, at least not to the extent that Mike D'Antoni pushed him last season. Misuse of the 7-footer's talents are more to blame for his 2012-13 struggles—career worsts 13.7 points per game, 46.6 field-goal percentage and 16.7 player efficiency rating—than any issues with aging.
As a secondary star for Rose, the Spaniard could give Chicago a steady hand in the low post. He's still a lethal shooter around the basket (54.4 percent from inside of 10 feet in 2012-13) and a supremely talented passer (career-best 4.1 assists per game last season).
He doesn't have Taj Gibson's ceiling defensively, but he doesn't have Carlos Boozer's basement either. Joakim Noah could handle the toughest post matchups, leaving Gasol in situations he'll win more often than not.
Kobe Bryant will fight to keep Gasol in L.A., but Rose should battle even harder to pry him out. Skilled 7-footers with championship experience don't come around often.
Rudy Gay has the chance to do something special next season.
No, he won't be leading the Toronto Raptors anywhere beyond the opening round of the playoffs (if he takes them that far). But he could climb back to a position of prominence, just in time for a round of contract negotiations, no less assuming he opts out of the final year of his deal.
With good size (6'9") and athleticism, Gay seemed headed for greatness as the No. 8 pick of the 2006 draft. Despite averaging better than 18 points in each of the last six seasons, the 27-year-old has left the analytical crowd wanting more.
His field-goal percentage bottomed out at last season's 41.6 mark, and his three-point success rate has held above 33 percent just once in the last four seasons.
So where should any optimism come from? More importantly, why should Rose's Bulls be interested?
Because Gay has taken steps to make sure 2013-14 is a career year. He had surgery to correct what he called "terrible" vision problems this summer, via Adam Figman of SLAM. He also found time to work with Hall of Famer Hakeem Olajuwon to fine-tune an offensive post game that's been productive but underused.
Moving from Deng to Gay might seem like lateral movement at first, but their ceilings are dramatically different.
Gay's problem has never been a lack of talent, it's just finding a way to maximize his natural gifts.
Thibodeau won't allow for a wavering defensive focus. With the right coaching, Gay could transform from a defensive afterthought to a suffocating stopper.
And with Rose guiding the offense, Gay should lose quantity but add quality to his offensive chances.
Even if the sides fail to reach an agreement by the Oct. 31 deadline, Utah will still retain his rights next summer and be able to match any offers made.
With the losses of Al Jefferson and Paul Millsap this summer and the addition of more than $24 million of expiring contracts, the Jazz are clearly in rebuilding mode. Hayward, a three-year veteran, figures to be a part of that plan.
But what would happen if the former Butler Bulldog gets overexposed in 2013-14? What if his unproven supporting cast forces him to take on more than he can handle?
It can't be ruled out as a possibility. As his offensive responsibilities have increased over the last three seasons, his field-goal percentage has plummeted. He was a 48.5 percent shooter as a rookie, 45.6 percent as a sophomore and just 43.5 percent in 2012-13.
Presumably the Bulls would have to overpay to get Hayward out of Salt Lake City, but maybe not if the Jazz go casting for bigger fish.
A move to Chicago wouldn't faze Hayward. He already plays with a certain Thibodeau style, displaying non-stop hustle and two-way energy. He could be a spot-up shooter (career 40.1 three-point percentage) for Rose or a setup man to move Rose off the ball in spurts.
If Hayward leaves Utah—and that's an admittedly large if—Chicago should be watching his every move.
Either Joe Dumars has a master plan to solve the Detroit Pistons' unavoidable spacing issues next season, or he spent as much time mapping out his frontcourt as he did weighing the benefits of investing nearly nine figures in Ben Gordon and Charlie Villanueva in 2009.
Maybe it's just me, but I'm not buying Brandon Jennings as the tactical floor general to crack this logistical puzzle.
Monroe's agent, David Falk, is already on record as saying that he won't look for an extension before next summer, via Patrick Hayes of PistonPowered.com. By that time this peculiar post pairing could look like a more jumbled mess in action than it already does on paper.
Monroe's one of the few bigs that could match Pau Gasol's passing ability (3.5 assists per game in 2012-13). At 23 years old, he has the time to bolster an already strong scoring base (16.0 points per game last season).
His defense needs a lot of work, but the Bulls could be his ultimate training partners. He'll get the easier defensive matchup while sharing the floor with either Joakim Noah or Taj Gibson, with the hope that one day he'll be demanding those assignments.
Monroe wouldn't be the best scorer that Rose has played with yet, but he could certainly grow into that role.
As long as LeBron James stays with the Miami Heat, Dwyane Wade will follow suit.
You don't just walk away from the chance to play alongside the world's greatest after all.
But the King remains clueless about his plans for 2014, via ESPN's Chris Broussard, so Wade's future is far from settled.
The Chicago native could be the perfect partner for Rose. He's still a superstar (21.2 points, 5.1 assists and 5.0 rebounds in 2012-13), but he's held his ego in check while ceding the spotlight to James.
There's certainly risk involved for any of Wade's suitors. He's already 31 years old with increasingly alarming knee issues.
But that's what should make a Windy City return look so appealing. He doesn't have to be the man with this franchise, maybe not even the second option if Joakim Noah continues to develop.
While neither Wade (career 28.9 three-point percentage) nor Rose (31.0) shoots the long ball with any consistency, their defensive ability would more than compensate for any offensive limitations.
A prime performer in nearly every facet of the game—defending, rebounding, scoring, distributing, hustling—Wade's an ideal addition for any roster.
With his championship background and suffocating defense, he may be as good as it gets for realistic future teammates for Rose.