According to Rob Pollard in the Manchester Evening News, the United hierarchy are prepared to shell out "close to £40 million to lure him to Old Trafford," but he has also been a long-term target for Chelsea.
United's defence has made a big improvement this season—they have conceded the second fewest goals in the Premier League—but they are still lacking a consistent centre-back pairing.
Chris Smalling has made a big step up in the current campaign, but his most regular partner, Daley Blind, is playing out of position, and 28-year-old Bonucci would be much more reliable.
Bayern Munich manager Pep Guardiola recently named the Italian centre-back one of his "all-time favourite players," a huge testament to his consistent quality, per UEFA.
He has been a rock in the Juve back line for some time now and has been almost ever-present in the Old Lady's current Serie A run of eight games without conceding a single goal, per WhoScored.com.
Bonucci also has a knack of scoring spectacular goals, as he did in Sunday's 2-0 win over Inter Milan, per Eleven SportsNetwork:
Eleven Sports BE @ElevenBE_fr
GOAL! Juventus 1 - 0 Inter. Bonucci. Live on Eleven Sports! https://t.co/EwUnxv9qij2016-2-28 20:56:33
And they often prove vital, per Opta:
9 - Bonucci has netted his ninth Serie A goal for Juventus: when the defender has scored, the Bianconeri have always won the match. Decisive2016-2-29 14:52:57
Despite Smalling's impressive form this term, United still need a seasoned leader in their defence, and Bonucci would add experience and quality in spades.
Juve will be loath to let him go, but they could well be tempted if the Red Devils are indeed prepared to spend big money on him.
It looks as though United will also have to spend big if they want to tempt Real Madrid into parting with James Rodriguez.
The Colombian has had form and attitude issues aplenty this season, and it has been widely reported that Los Blancos could be prepared to let him go in the summer, with United an interested party, per the Sun.
James was again off form as Real lost to Atletico Madrid at the weekend, prompting Bleacher Report's Dean Jones to suggest he will now be sold:
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However, while Sergio Santos Chozas of AS reported Real "are considering selling" James, 24, in the summer, he added "only an absurd offer would see the 2014 World Cup hero departing the Spanish capital before the start of the 2016/17 campaign."
The rationale offered for holding onto James is that Real believe manager Zinedine Zidane can get the best out of him, and it would be a huge risk on United's part to table a massive offer for the playmaker given his current troubles.
Elsewhere, United have been given a slight boost in their reported pursuit of Athletic Club Bilbao centre-back Aymeric Laporte.
Sky Sports Spanish football expert Guillem Balague believes Barcelona may not have the money to enter the race for the Frenchman this summer, meaning one fewer major club for the Red Devils to compete against.