The Geordie is also wanted by Manchester City, though The Sun says United are willing to go the extra mile to beat their local rivals.
Anderson has a deal with Forest to 2029, though there is an acceptance at the City Ground that he will be on the move this summer.
Interestingly, Anderson shares the same agents - Stellar - as United midfielder Kobbie Mainoo, who is now set to sign a new contract.
United management have plans to sign two midfielders over the summer.
Anderson, Adam Wharton (Crystal Palace), Sandro Tonali (Newcastle), Carlos Baleba (Brighton) and Alex Scott (Bournemouth) are among the Premier League midfielders being tracked by United this season.
