The midfielder scored as United progressed to the Europa League quarter-finals by beating LASK but was left frustrated after being substituted after 63 minutes and Scholes believes he may not have the quality to stay at Old Trafford.
Speaking on BT Sport, the legendary former midfielder said: "He is capable of more than he's showing at United.
"When he's playing well he has something to offer but lately he has not been good enough. Is he a starting XI player if you're going to win the league and big trophies at Manchester United? I'm not sure.
"It was a good chance for him to play 90 minutes, but when Ole [Gunnar Solskjaer] took him off it is a sign Ole is thinking of moving him on."
Solskjaer's side look set to land Borussia Dortmund star Jadon Sancho in the transfer window and Scholes believes that Lingard may have to step down a level if he wants regular game time.
He said: "I know what Jesse is capable of, I grew up with him, played with him, coached him in the reserves, but he has just become a squad player.
"He should be at the peak of his career and does he only want to play 10/15/20 games?
"Jesse is a good lad, he just needs a run of games but I don't think he will get that at Manchester United, maybe at a mid-table team like an Everton or West Ham."