The Germany international, whose contract expires in 18 months, is expected to leave Borussia this summer amid strong interest from both United and Madrid.
United manager David Moyes has travelled to Spain, Italy and France to scout players this month and will be present for Napoli's Coppa Italia quarter-final with Atalanta on Wednesday night. Any deal for Gündoğan is likely to be concluded in the summer, though.
Premier League champions United were linked with Gündoğan as far back as three years ago, when he was at Nürnberg. The 23-year-old has also hovered onto Madrid's radar since his exceptional performance in Dortmund's 4-1 thrashing of Real in last year's Champions League semi-final.
Gündoğan, who has been offered a new contract by Borussia, told Bild: "I will soon make a decision that will be well considered. It's still really open.
"I'm trying to be fit and to get back onto the pitch. This is important and a top priority for BVB and for me."
Although Gündoğan could become the third influential player to leave Jürgen Klopp's squad after Mario Götze and Robert Lewandowski, he is at least unlikely to join Bayern Munich.
Dortmund will be particularly keen to avoid a repeat of the Lewandowski debacle, after the Poland striker signed a pre-contract agreement with Bayern earlier this month to join them on a free transfer in the summer. Götze's €37.5 million buy-out clause was triggered by Bayern in April.
Klopp has at least had time to experiment without Gündoğan, sidelined since August through injury. Nuri Şahin, who could rejoin the Yellow and Blacks on a permanent deal from Real this summer, has occupied the creative central midfield berth alongside Sven Bender.
The Şahin situation possibly gives Madrid the advantage over United should they negotiate for Gündoğan, while it is also rumoured they will offer Karim Benzema as part of any deal.
Dortmund would want over £20m for Gündoğan, although it is possible United could use the under-performing Shinji Kagawa, who excelled at Borussia, as a makeweight.