A United side showing 10 changes from Monday's victory at Aston Villa fell to a 2-1 loss at home to Leicester City on Wednesday (AEST) - a result that crowned City as champion with three games to spare.
It is City's seventh top-flight title triumph, five of those in the last decade, while United's wait for Premier League glory will stretch into a ninth season.
While Solskjaer feels his side deserve praise for pushing City as far as they did, the Norwegian accepts it will take some spending in the close season to fully close the gap.
"If you want to take up the challenge with them we need to strengthen our squad," he told BT Sport. "Congratulations to them. They have had a fantastic season.
"I am very pleased with my boys that have made it to the last 10 days with a Manchester City side ahead of us that are lauded as the best in Europe.
"We want to take the next step and it might be that we need a couple more to strengthen the squad."
Mason Greenwood was the only player to keep his place in United's starting line-up for the loss to third-placed Leicester, who are now within four points of their opponents.
Solskjaer hinted ahead of the game he would prioritise the upcoming clash with Liverpool, which comes a little over 48 hours after the visit of Leicester to Old Trafford.
The United boss believes his side performed well given the circumstances as he again hit out at the Premier League for rescheduling the game versus Liverpool for this Thursday.
"We started off slow, as you might expect because they have not had any time to prepare for the game, but after they scored we had nothing to lose," he said.
"We played some very good stuff and I was pleased with it. We have not had too much time to prepare and we went into half-time pleased.
"They didn't get out of the traps and they got a goal from a corner that should have been a throw.
"But it can't be like this again. We can't play Thursday, Sunday, Tuesday to Thursday. It is impossible. That's the reason we made the changes.
"It's unheard of. It is impossible for the players to perform at this level going from Thursday, Sunday, Tuesday, Thursday."
Greenwood's smart finish cancelled out Luke Thomas' impressive volleyed opener, before Caglar Soyuncu headed in a winner for Leicester 24 minutes from time.
England international Greenwood now has 29 goals in all competitions since the start of last season, which is the most of any teenager in Europe's top five leagues.
On a day that saw Amad Diallo and Anthony Elanga make their Premier League debuts for United, Solskjaer reserved special praise for Greenwood.
"Mason looked like a seasoned professional and he is only 19," Solskjaer said at his post-match news conference.
"You saw some definite leadership in he had to take responsibility. He did really well considering his age.
"No matter how old you are, if you believe yourself and trust yourself then you make that decision."
The defeat for United was their fifth at Old Trafford in the Premier League this season, which is as many as in their previous two campaigns combined.