The Germany international has been superb for Dortmund since joining from FC Nuremberg in 2011, and played a key role as Jurgen Klopp's side finished last season as Bundesliga champions.
His composed performances at the heart of the Dortmund midfield are believed to have alerted Manchester United boss Sir Alex Ferguson, although Spurs manager Andre Villas-Boas is also thought to be tracking the 21-year-old.
The report claims that Ferguson is monitoring Gundogan as he continues to assess Borussia Dortmund striker Robert Lewandowski.
The Polish international has emerged as a transfer target for a number of clubs, but failed to impress as Poland drew 1-1 with England last week.
Gundogan has been rated at £20m, a price which could go higher if the Premier League pair enter a bidding war when the transfer window reopens in January.
United signed another former-Dortmund star over the summer, spending £17m to bring Japan international Shinji Kagawa to Old Trafford.
Dortmund are struggling in the Bundesliga as they look to defend their title, as defeat to Schalke at the weekend allowed Bayern Munich to extend their early lead, and it doesn't get any easier as Klopp's side face Real Madrid in the Champions League this week.