The Red Devils are believed to have identified Zirkzee as their number one centre-forward target, with the club looking to bring in competition for Rasmus Hojlund for the 2024-25 campaign.
Zirkzee has a €40m (£33.9m) release clause in his current contract, which Man United have allegedly decided to activate.
Arsenal and AC Milan have also been linked with the Dutchman, who appears to have played his final match for Bologna, with the Italian outfit expecting to lose him this summer.
According to Corriere di Bologna, Man United have now reached an agreement with Zirkzee over personal terms, with a deal potentially being completed in the near future.
Man United 'closing in' on Zirkzee deal
The report claims that Zirkzee had reservations over a move to Old Trafford, as he was unclear how much football he would receive due to the presence of Hojlund.
However, Ten Hag is said to have played a major role in convincing the 23-year-old that he would be an important player for the Red Devils if he joined the club.
Man United's head coach is believed to have directly contacted Zirkzee to discuss his role in the team, which convinced the striker that the English giants should be his next club.
Hojlund scored 16 times for Man United last season after arriving from Atalanta BC in the summer of 2023.
However, Anthony Martial's departure on a free transfer means that the Denmark international is currently the only senior centre-forward available for selection.
Zirkzee had a productive 2023-24 campaign at Bologna
Zirkzee was in strong form for Bologna during the 2023-24 season, scoring 12 goals and registering seven assists in 37 appearances in all competitions.
The striker has now managed 14 goals and nine assists in 58 appearances for his current side, while he also had a successful loan spell at Anderlecht earlier in his career, netting 18 times and registering 13 assists in 47 matches.
Zirkzee started his professional career at Bayern Munich, and he managed four goals and one assist in 17 appearances for the German giants before moving on.
The forward market is a difficult one this summer, and for €40m (£33.9m), Zirkzee could turn out to be a bargain for Man United considering his potential.
Ten Hag's side are also believed to be closing in on a deal for Bayern's Matthijs de Ligt, with the Netherlands international potentially being confirmed as the English club's first signing of the summer in the not too distant future.