The 24-year-old German has just one year remaining on his current Dortmund contract and is believed to be keen on a move to the Premier League, with three of the countries top sides interested in securing his services.
Gundogan is valued at £20 million, and City believe they can steal a march on their rivals by offering misfiring striker Edin Dzeko to Dortmund in a swap deal.
New Dortmund head coach Thomas Tuchel, who will take over from the outgoing Jurgen Klopp this summer, is an admirer of Dzeko’s and that could see Gundogan seal a move to City.
Arsenal are known to have had long-term interest in the German midfielder who has nine caps for his country, but could be beaten to his signature by one of the Manchester clubs.
United are closing in on a £21.5 million deal, according to Sky Sports, as the race for Gundogan looks set to heat up with the season drawing to a close.
He has struggled with injuries in the past few years but has made 27 appearances in all competitions this season, scoring three goals with four assists.
Klopp is being strongly linked with the Manchester City job and if the Dortmund coach is to move to the Etihad this summer, Gundogan will be expected to follow him.
Where will Ilkay Gundogan move this summer? Who needs him most – Manchester City, Manchester United or Arsenal?