United have now agreed a £40.3 million deal with the Dutch club for the 23-year-old and a move could be finalised this week.
United also remain intent on signing Jadon Sancho, however they have still been unable to agree a transfer fee with Borussia Dortmund.
Solskjaer is also looking to bolster his defensive options ahead of the new season and according to ESPN, the United manager is pushing to sign Upamecano from RB Leipzig.
The report claims that Solskjaer is ‘prioritising’ a new central defender to play alongside Harry Maguire, while United believe an agreement can be reached with RB Leipzig before the transfer deadline on October 5.
In July, Upamecano signed a new contract with RB Leipzig which keeps him tied to the club until 2023.
However, reports in Germany claim that the 21-year-old’s new deal includes a €42m (£37.6m) release clause which will be active from next summer.
United, meanwhile, are open to offloading Chris Smalling, Phil Jones and Marcos Rojo before the end of the transfer window.
ESPN also reports that Solskjaer is also ‘concerned’ by the reliability of Victor Lindelof and Eric Bailly, which is why he is keen to recruit a new centre-back for United this summer.
‘I think they could do with a centre half, I’ve said that all along,’ said Scholes.
‘I’m not sure Lindelof going to be good enough as a partner for Maguire. I think they need a dominant defender next to him.
‘Defensively, the full backs are OK. On the ball you’d have to question but defensively they’re OK.
‘I think the defensive record over the last 15, 20 games has been very good, probably one of the best in the league so you have to take that into respect as well.
‘But I still think a dominant centre-half next to Maguire, just to cover for the slight weaknesses he has.’