Whether United stump up the cash to bring him to Old Trafford remains to be seen, especially with the club also having other targets in mind.
A right-back is being sought to help spur Aaron Wan-Bissaka on, with Diogo Dalot unlikely to return once his loan spell with AC Milan comes to an end.
United retain interest in Borussia Dortmund’s Jadon Sancho, despite seeing a move for the England international break down back in the summer.
And Sancho’s team-mate, Erling Haaland, continues to be linked.
Meanwhile, speaking recently, former Liverpool star Peter Crouch expressed a belief that Rice would be a good signing for United as they eye the Premier League title.