Gundogan has been linked with a move away from Dortmund all summer and at one point it seemed like the German international is destined for Bayern Munich.
Gundogan is also a target for Juventus and would be open to making a move this summer.
The German midfielder has struggled with injury issues in 2014, but is looking to get back to his best this season.
Gundogan has been impressive under Thomas Tuchel so far this season and ended last season strongly as well.
After the departures of Pirlo and Vidal, Allegri is keen to sign the German to strengthen his midfield.
Elsewhere, Manchester United are also looking to sign a creative central midfielder and have tabled an offer for Gundogan.
Although Dortmund are insistent that none of their players will leave this summer, it will be interesting to see what happens if Gundogan decides to force a move.
Gundogan is not the only player in Bundesliga being courted by Manchester United. The Red Devils are also looking to sign Thomas Muller from Bayern Munich this week and are ready to offer 80 million euros for him.
Gundogan turned down a contract offer at Dortmund previously and could look to force the move if United’s interest is serious.