The Belgium international has spent the last four seasons with Borussia Dortmund, making a total of 145 appearances in all competitions.
However, Witsel, who turned 33 at the start of the year, is departing the Westfalenstadion on a free transfer this summer with his contract not being renewed.
According to Der Westen, United and Spurs are contemplating whether to put forward a proposal for the veteran midfielder as they bid to strengthen their options in the middle of the park.
As far as United are concerned, Witsel may prove to be a replacement for Nemanja Matic, who has already said that he will be leaving Old Trafford.
Spurs head coach Antonio Conte is also looking for competition for the likes of Pierre-Emile Hojbjerg and Rodrigo Bentancur, and the pull of Champions League football may prove key with Spurs requiring just a point from their final Premier League fixture at Norwich City.
Juventus are also said to be monitoring the situation as they look to shake up their own midfield. body check tags ::