The Sun says Palace boss Frank de Boer wants to reunite with Fosu-Mensah, having worked with the 19-year-old previously when he was manager of Ajax.
Palace have also put together a £20m ($26m) package to sign Calum Chambers from Arsenal, but De Boer wants to see if United are willing to let Fosu-Mensah leave on loan before signing off on a move for Chambers.
The Eagles manager wants to bring in a right-sided defender to play in a 3-4-3 formation alongside Scott Dann and Jairo Riedewald.
Fosu-Mensah featured in five of United's seven games in pre-season, mostly as a right-sided full-back.
The Dutchman, who has also been linked with a loan move Watford, said after the club's 3-0 win over Valerenga that he was looking forward to the clash with Real Madrid in Macedonia on 8 August.
"Everyone got their minutes tonight, so I think the whole team is building up to fitness and it's going really well," Fosu-Mensah told MUTV. "We've won a few games and we're working towards the game against Real Madrid next week.
"I think we're stronger than we were last season. We're working on a couple of points and we've had a couple of great additions to the squad in Romelu [Lukaku] and Victor [Lindelof]. I think we're stronger.
"That's what we're going for [the Premier League title]. We're Manchester United, so of course that's what we are going for."
Fosu-Mensah added that he did not know if he would serve as Antonio Valencia's deputy at Old Trafford in the upcoming season.
"I'm just trying to develop my game. I'm young and still have a lot to learn, but I'm learning every day," he said.
"I'm working with one of the best managers in the world, it's something I'm grateful for and, as I say, I'm learning every day."