The Manchester United duo have been tipped to quit Old Trafford this summer and head to Spain.
Pogba has publicly discussed the idea of joining Real Madrid, with the France international claiming it would be a dream to head to the Bernabeu.
Rashford, meanwhile, is out of contract next summer – and Barcelona are tracking the England star.
Both Pogba and Rashford have been instrumental under Solskjaer, with the United duo a key part of the Norwegian landing the job full time.
And Solskjaer knows United cannot afford to lose their stars this summer if they are to challenge for the Premier League title next season.
When asked if both Pogba and Rashford can win the trophies they want to at United, Solskjaer said: “I would believe so.
“That’s our aim as a club – to be achieving what we have done in the past and that’s the expectation here.
“And, of course, we’re hoping to build a team strong enough and worthy of the history of the club.
“You’ve got loads of players who you can build a team around and those [Pogba and Rashford] are the two that you asked me about. They’re vital for our future.”
Rashford and United are locked in talks over a new contract.
The striker, 21, could see his wage rocket from £75,000-per-week to around £200,000-a-week with add-ons.
Rashford is likely to extend his United contract, especially following the permanent appointment of Solskjaer this week.
However, Pogba’s future is up in the air.
The World Cup winner has two years remaining on his current Old Trafford deal, however, the midfielder has talked up a move to Real Madrid.
Pogba revealed he’s happy under Solskjaer at United, however, the France star admitted it would be a dream to head to the Bernabeu.
“Real Madrid is one of the biggest clubs in the world,” Pogba said.
“I’ve always said it is a dream club for every player.
“At the moment I’m happy in Manchester. I play. There’s a new coach.”
Solskjaer urged his United stars to continue their fine form and challenge for the Premier League title next season.
The Red Devils are back in action on Saturday afternoon, with Watford travelling to Old Trafford (3pm kick off).
A win for United could send them into the top four ahead of Arsenal.