United agreed a initial £34m deal to sign Varane, who has entered the final year of his contract at Madrid, with the agreement potentially rising to £42m with add-ons.
The 28-year-old is regarded as one of European football's leading centre-backs and will join up with fellow new arrival Jadon Sancho, who completed a £73m move from Borussia Dortmund earlier this month.
Varane is expected to partner Harry Maguire in the centre of Solskjaer's defence, with Victor Lindelof and Eric Bailly providing cover, but such strength in depth in the centre of defence opens up the possibility of United playing a back three.
After the 2-2 pre-season friendly draw with Brentford at Old Trafford on Wednesday evening, Solskjaer hinted that Varane's imminent arrival has tempted him to consider different systems and set-ups ahead of the new season.
“I'm very happy getting both [Sancho and Varane] in," Solskjaer said.
“First of all the club shows ambition with one of the most exciting young forwards in world football and one of the most respected centre-backs in the world and one of the most winning centre-backs in the world.
"With him in, we've got different ways of playing as well, I can see different ways of playing with Raphael," he added. “I can't wait to get him in, hopefully we can get that sorted as soon as.
“He's a proven winner, a player we've followed for many, many years. I know Sir Alex [Ferguson] was very, very close to signing him.
“This time around, we've got him at the other end of his career. He's proved over his whole career what a good person he is, what a professional he is and he's won all that there is to win."
United moved early in the summer window to secure both Sancho and Varane, who were Solskjaer's two priority transfer targets after last season's finish as distant runners-up to Premier League champions Manchester City.
Solskjaer believes that completing two major pieces of transfer business early has left United in a strong position, with just over a fortnight until the top flight restart, and expects both major new signings to make a "huge difference".
“I think as I've said before, every manager would be happy the sooner and the earlier the deals are done," he said. "This year we've managed to get it sorted. The two we've got in will make a huge difference for us, this season short-term and also long-term.
"Raphael is in his prime, he's the same age as Harry, a year older than Eric and a couple years older than Victor. I've got a fantastic group of centre-backs.
“With Jadon as well, with the young forwards we've got it's exciting. As a manager you can't ask for more backing when you've got these two in as early as we have.
“We've scrambled before towards end of the window and now I have to say I think we're in a good position.”
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