Raphael Varane and Jules Kounde are both admired, with Ole Gunnar Solskjaer hoping to sign a new centre-back to go alongside Harry Maguire.
At right-back, Kieran Trippier and Max Aarons remain under consideration amid their fine showings for Atletico Madrid and Norwich City.
And, up top, United want to sign Erling Haaland – regardless of what happens with Edinson Cavani.
Rice, however, has now emerged as a target in central midfield.
And The Athletic say that, during the last international break, the West Ham vice-captain had ‘light-hearted’ talks with the United players in the England camp.
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Solskjaer is ready to explore the possibility of signing the Hammers star during negotiations for Jesse Lingard, who has taken the London Stadium by storm since his January arrival.
But any deal may be tricky to strike, with West Ham previously insisting they’ll demand over £100million to let Rice leave the club.
United have been impressed by Lingard’s form at the club so far, with the England international scoring six goals and registering two assists from eight showings in the capital.
They believe this turnaround has been sparked by David Moyes’ willingness to give the 28-year-old more responsibility, freedom and a license to try things.
And whether United are successful in signing Rice remains to be seen.
Solskjaer has tried to sign some of the best young players in the top flight throughout his reign as manager so far.
After seeing Antonio Valencia depart in 2019, the Norwegian sanctioned a huge £50million deal for Aaron Wan-Bissaka.
They also spent big on Harry Maguire, forking out £80m in order to bring the England international to Old Trafford that same year.
And, had all gone to plan, United would have added Jack Grealish to their ranks last summer – only for the coronavirus crisis to ultimately scupper a deal.
Speaking last month, Owen Hargreaves tipped Rice to move to Old Trafford – and thinks his potential is ‘off the charts’.
“I think Rice’s upside is off the charts,” he said.
“He has the potential to be anything and a lot better… I think he plays within himself at West Ham.
“People don’t realise how big he is… he’s 6ft 3in, so he looks like a centre back but he’s actually really good technically.
“I’m not a big fan of players who just play in there and pass it around, you have to stop players too.
“Declan Rice doesn’t have to dictate the game for Man Utd, he can pass to the players who can make the difference and be good defensively.”
Moyes said in March that Rice was worth more than £100m in today’s world.
“I don’t agree with the owners that Declan Rice is a £100m footballer. Far, far more than £100m. Far, far more,” said the Scot.
“I look at the players who have joined some clubs from overseas and come in and nowhere near had the impact Declan Rice would have.
“I certainly wouldn’t put a price on it. You can decide which figure you want to put on it but I wouldn’t be sanctioning anything like that, that’s for sure.”
Moyes also played down talk that West Ham would need Champions League football to keep Rice at the London Stadium.
He added: “Do we need the Champions League to keep Declan? No, because Declan is under contract first and foremost, so he can’t go anywhere no matter what, and I keep saying it and I hope people know that when I say something, I mean it.
“We’ve not had an offer for Declan Rice and I hope we don’t get one.
“And you’ve got an idea now of where we’d need to be if it was even going to be considered, certainly be me.
“If we could finish in Europe, in some sort of European position, I think it would be progress.
“And that would be progress for Declan Rice as well because he’s been at West Ham while they’ve been bobbing around the bottom, so this is new for Declan as well.”