The Red Devils have shown intent this off-season landing experienced French defender Varane along with Borussia Dortmund winger Jadon Sancho.
The 28-year-old made more than 350 appearances for Real, winning numerous titles including three La Liga crowns and the UEFA Champions League four times.
“Of course, I’m very happy that we’ve managed to get the deal with Real Madrid,” Solskjaer said.
“He’s a proven winner and a player we’ve followed for many, many years. I know Sir Alex was very, very close to signing him and 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, first of all. Professional and he’s won all there is to win.”
Solskjaer was also delighted with United’s off-season transfer activity as they look to improve on last season’s runners-up finish in the Premier League.
“I’m very happy getting both our players in,” he said.
“First of all, the club has shown ambition with getting one of the most exciting young forwards in world football (Sancho), and one of the most respected centre-backs and one of the most winning centre-backs in the world.
“With him in, we’ve got different ways of playing of course, and I can see different ways of playing with Raphael. I can’t wait to get him in and hopefully we can get that sorted as soon as.”