Manchester United boss David Moyes is reportedly eyeing a host of high-profile targets ahead of the January transfer window, with Gundogan believed to be high on his list of priorities alongside Andres Iniesta and Marco Reus.
However according to the Daily Mail, Real Madrid are increasingly confident of landing the midfielder, with club officials rubber-stamping the terms they are willing to offer him.
The paper adds that Los Blancos are willing to offer Borussia Dortmund £25 million for his services.
With the Mail also reporting that Manchester United look set to be unsuccessful with their bid to land his Dortmund star Marco Reus, they could be facing another frustrating transfer window after failing to complete a string of high-profile deals in the summer.
Eventually Moyes landed Marouane Fellaini from Everton on deadline day for £27.5 million, but not before he had tried and failed to snap up the likes of Sami Khedira, Cesc Fabregas, Thiago Alcantara, Fabio Coentrao and even Gareth Bale.
The Mail report that Manchester United are still keen on Coentrao - as are Tottenham - and are also monitoring Sevilla’s Alberto Moreno.
The Red Devils had a shaky start to the new season under Moyes, who was named as Sir Alex Ferguson's successor at the end of the 2012/13 campaign, although their form has steadily improved and they now sit 5th after 11 games following their 1-0 win over league leaders Arsenal on Sunday.