Solskjaer has had a mixed time as United boss, enduring difficulties for much of this season but showing promise before the campaign came to a halt due to the coronavirus pandemic.
The Red Devils have been linked with a number of high-profile managers including Mauricio Pochettino and Max Allegri.
But Schmeichel claimed in an Instagram Q+A session that they are better off sticking with his former United teammate, with whom he won the treble alongside in 1999.
“I think Ole has been absolutely fantastic,” Schmeichel said. “Fingers crossed that everyone in the boardroom and the ownership circle see the same thing, that they’re happy with what got and don’t see the need to change to yet another big-name manager.
“We have a big-name manager. Ole Gunnar Solskjaer is one of the biggest names in Manchester United’s history, so we don’t need anybody else.
“Ole had them going really well. It was eleven games undefeated, before everything was closed down. It was a big shame, because the job that Ole’s done, it really is an amazing job.”
United could miss out on Champions League football for the second season on the trot as they trail fourth placed Chelsea by three points.
But Schmeichel claims the Norwegian is doing a good job of transforming the United squad, shipping out dead wood and adding new stars.
“I don’t think he was given the best squad of players to do precisely what he wanted to do,” he added. “He realised very quickly it was going to take time and he spent that time wisely.
“The players he knew for sure that had to go, I think most of them have gone. I think the players he brought in were good, in the sense of what he wanted to do.
“So I’m actually looking at United being very successful this season, with the introduction of new players and still in touching distance of Champions League qualification.”