The German international is widely regarded as one of Europe’s top talents and has been a key part of the Champions League runners-ups’ squad in recent seasons.
His fine displays have attracted interest from across Europe, with United and Arsenal just two sides believed to be keen.
After missing out on a number of targets over the summer, Moyes is ready to make a serious move for the 22-year-old.
Gundogan’s contract with Dortmund expires in 2015 and a lack of development regarding an extension has opened the door for a big move.
United were interested in the silky playmaker in 2011, with the player himself admitting that a move to Old Trafford would be a “dream”.
Arsenal are also keen, but Arsene Wenger’s need for striking reinforcements may rule him out of a bidding war for the youngster.
Gundogan is valued at around £30m, but may be available for a reduced fee if contract negotiations falter.
The ex-Nuremburg man is currently sidelined through injury.
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