The Milwaukee Brewers starter has just one year remaining on his current contract, making him an attractive target for playoff hopefuls. At the same time, the 2009 AL Cy Young winner is just 28 years old and could be a fixture at the top of the rotation for whatever team acquires him.
However, the price for Greinke could be steep. With the number of teams looking for pitching help, the Brewers will likely expect multiple blue-chip prospects from the highest bidder, and the price tag could drive teams away.
Ultimately, only one team will get to pay the price for Greinke. Here are the seven teams who have the best chance of landing one of the summer's biggest prizes.
The Atlanta Braves are in the thick of the NL wild card race, but their pitching leaves something to be desired.
Right now, Tim Hudson and Tommy Hanson have been mediocre at the top of the Braves rotation. A pitcher of Greinke's caliber would step in and immediately become the ace in Atlanta.
If the Braves do end up trading for Dempster, the Greinke issue becomes moot. If another team takes Dempster away from Atlanta, the organization still has the prospects to appease the Brewers.
He might not be the Braves' first choice, but Greinke could still find himself in Atlanta for its run at the playoffs.
One starter might not solve the Orioles' pitching woes.
Though the Boston Globe considered Baltimore a dark horse in the Greinke sweepstakes, the Orioles could use to acquire multiple pitchers if they want to remain competitive in the AL wild card race.
With a 3.54 ERA and 8.73 strikeouts per nine innings, Jason Hammel has been solid as Baltimore's de facto ace. Wei-Yen Chen also has a WAR over 2.0, giving the Orioles two decent starters at the top of their rotation.
The wheels fall off after that, though. Brian Matusz, Tommy Hunter, and Jake Arrieta all have ERAs over 5.00 and negative WARs. Greinke is just one man, and he cannot offset the production of three struggling pitchers.
He could certainly help in Baltimore, but Greinke isn't exactly what the Orioles need. Expect Baltimore to balk at sacrificing the future for a guy who can't save the Orioles in the present.
The Red Sox have bigger names in their rotation than the Orioles do. However, their numbers are demonstrably worse.
Josh Beckett and Felix Doubront are neck and neck for the rotation lead in ERA, with Beckett holding a 4.53 to 4.54 edge. In terms of WAR, Beckett leads the group again with a paltry 0.8, while Jon Lester brings up the rear with a minus-0.8.
As Boston's starters struggle to be mediocre, it is curious that pitching coach Bob McClure wouldn't want Greinke to replace any of them.
"I think we have some pretty good guys here," McClure told NESN, "but I wouldn't do it just in spite of one of our other guys because we have six guys here who can do a pretty good job. If you're just saying 'Would you do it?' as a front office person, I'd definitely have to think about it. It depends on what it'll cost you. It ain't gonna come cheap."
Though trading for Greinke isn't McClure's call to make, it seems like the acquisition wouldn't have the full support of the franchise. If McClure's faith is any indication, Boston will simply hope that their guys rebound for the final months of the season.
The Chicago White Sox already have a staunch starting rotation, but Greinke would strengthen their shot at the World Series.
Chris Sale and Jake Peavy were both named to the AL All-Star team this season. Sale is on fire with a 2.37 ERA and 4.4 WAR, and Peavy is not far behind with a 3.22 ERA and 3.6 WAR. Further down in the rotation, Jose Quintana has established himself as a good third starter with a 2.58 ERA and 2.8 WAR of his own.
With that group of pitchers, Greinke would be Chicago's third starter at best. Nonetheless, Ken Rosenthal and Jon Paul Morosi of Fox Sports expect the White Sox to make a strong push for Greinke. They predict Chicago will bring a third team to the table in order to build a better trade package built around Gavin Floyd.
It remains to be seen if Chicago can find the right facilitator to make this trade work. But the White Sox are hungry and looking to make a big deadline deal; all indications point towards Greinke being their guy.
The Angels are going to need some help if they want to succeed in the surprisingly competitive AL West.
Los Angeles currently holds one of the two AL wild card spots, but it is five games behind the division-leading Rangers and tied with the surging A's. Though Jered Weaver and C.J. Wilson have been stalwarts at the front end of the rotation, Dan Haren and Ervin Santana have each disappointed with negative WARs thus far.
CBS Sports speculates that in order to get Greinke, the Angels would have to give up Peter Bourjos, Garrett Richards, or both. Though neither has been a major contributor for the Angels this season, each has long-term potential. The team could be loathe to part with two of its best young players just to rent Greinke.
On the other hand, the Angels could use some reinforcements if they want to challenge the Rangers for a secure playoff spot. Don't be surprised if free-spending Los Angeles pulls the trigger to improve its rotation.
The Angels aren't the only team in town that's interested in Greinke.
Over in Chavez Ravine, the Dodgers have a surprisingly productive rotation already. Los Angeles has the reigning Cy Young winner in Clayton Kershaw and Chris Capuano has held his own with a 2.4 WAR as well. However, the Dodgers offense ranks 12th in the NL in runs scored, so they will have to rely on their pitching to win down the stretch.
Los Angeles is casting a wide net in its search for more starting pitching. In addition to Grienke, Sporting News reports the Dodgers are looking into Dempster, Matt Garza, and Cole Hamels. They have also been rumored to be pursuing offensive firepower, which would divert resources away from a trade for Grienke.
Though Greinke could power the Dodgers to an NL West title, the team is exploring too many options to believe it would give Milwaukee the best offer.
Due to Colby Lewis' torn flexor tendon, the Rangers have a hole to fill in their rotation. Unlike the other teams on this list, Texas currently has starters to spare.
Matt Harrison and Yu Darvish are enjoying good seasons, while Derek Holland and Roy Oswalt have left something to be desired. With Lewis out for the season, the Rangers could turn to Scott Feldman or Alexei Ogando. However, Feldman has a 5.37 ERA on the year, and Texas would prefer to keep Ogando in the bullpen.
All indications point to Texas reloading its rotation through a trade. The team's website reports the Rangers want to add a "difference-maker" to the rotation, with Greinke and Hamels leading their wish list. Texas has coveted prospects in Jurickson Profar and Mike Olt, and it would not hesitate to move them for a pitcher like Greinke.
The Rangers have their sites set on the World Series this season. If anyone is going to pull out all the stops to get Greinke, it's them.