United have agreed deals for Sancho and Varane for a combined fee of more than £100 million and Solskjaer admits he "could not have asked for more backing" as he attempts to build a squad capable of challenging for the Premier League title.
"I'm very happy that we've managed to get a deal with Real Madrid [for Varane]", Solskjaer said after the 2-2 draw with Brentford on Wednesday night. "He's a proven winner and a player we've followed for many, many years.
"The club has shown its ambition with one of the most exciting young forwards in world football [Sancho] and one of the most respected centre-backs in the world.
"As I've said before, every manager would be happy the sooner the better the deals are done and this year we've managed to get it sorted. The two we've got in will make a huge difference for us and as a manager you can't ask for more backing when you've got these two in as early as we have."
With the imminent arrival of Varane, Solskjaer says Axel Tuanezbe, wanted by Newcastle, could go out on loan. Phil Jones, meanwhile, who has not played a competitive game for 18 months because of a knee injury is expected back in training this weekend.
"With Raphael looking like everything is sorted there, I can see Axel going out because I've got Raphael Varane, [Harry] Maguire, [Victor] Lindelof and Eric Bailly," said Solskjaer.
"Phil Jones is due to start training over the weekend when we've got everyone in on Monday. It's a big plus for him. I know exactly the feeling of being out for so long, he's always served the club and it was time to look after him and his own health."
United's remaining internationals, including Sancho, are due to return to training on Monday and Solskjaer says a decision will be made next week on whether Marcus Rashford will undergo shoulder surgery.
"We'll probably make a decision in the next few days when we check him up," said Solskjaer. "What's best for him and what's best for the club has to be considered."