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Solskjaer is reportedly set to receive a new lucrative three-year deal with his current deal due to expire in June 2022.
And Scholes has backed his former side's decision to hand him a bumper new deal.
However, he believes the 48-year-old will have to deliver some silverware soon and calls United's recent run of form in semi-finals as worrying.
"I think he probably deserves it. There's been progress from the manager before [Jose Mourinho]," he said on the Webby & O'Neill YouTube show.
"I think the team looks a lot more exciting, there are goals in the team, which you would expect from a United team anyway.
"I think the only problem is: winning something. Losing four semi-finals is a little bit of a worry and losing that quarter-final to Leicester [in the FA Cup], it wasn't good for fans especially.
"The Europa League was tough in Milan, I thought players looked a little bit jaded - [Bruno] Fernandes especially, he did need the rest against Leicester.
"But on the other hand, I don't think United are in a position to rest people for the big games.
"They have got 12, 13 players that are good enough to go and win a lot of games. But is the squad then deep enough when you've leaving these big players out to go and win the bigger games?"
The United legend added: "Out of the FA Cup and the Europa League, I'd still want to win the FA Cup. The Europa League is not where United want to be or should be.
"But they're in it and they have a great chance of winning it with the teams are left in it. I think there is progress but I just think Ole needs to try and win something."
United are favourites to win the Europa League this season having beaten AC Milan in the last round.
It is their only chance of a major trophy this campaign - but if they fail to win it, it will mean a fourth consecutive term without silverware.
The Red Devils, who are currently second in the league, are on course for a top four finish which would mean Champions League qualification.
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