Solskjaer, appointed manager on a full-time basis on Thursday, has set himself the target of bringing the Premier League title back to Old Trafford for the first time since 2013.
The Norwegian says there will be ins and outs this summer as he looks to build a team capable of challenging Manchester City and Liverpool.
And, despite interest from Real Madrid and Barcelona, the 46-year-old said Pogba and Rashford are central to his plans.
"They are vital for our future," said Solskjaer. "Our aim as a club is to be achieving what we have done in the past.
"That is the expectation here and of course we are hoping to build a team strong enough worthy of the history of the club.
"You have got loads of players who you can build a team around."
At his official unveiling on Thursday, the new permanent boss was clear that he will spend money in the summer to supplement his squad but he has assured executive vice-chairman Ed Woodward and the Glazers that he will not go wild in the transfer market.
"United is a club that has money, and that has to be spent in a reasonable way," he said. "People who know me, know that I will use that in a good way. We have a plan with the club, they have scouted players for longer than I have been here, and we have players that we need to keep.
"We will not spend money just to spend money."
Solskjaer has been rewarded with three-year contract after winning 14 of his 19 games in charge during a successful caretaker stint.
Watford's visit to Old Trafford will be Solskjaer's first since being confirmed as Jose Mourinho's permanent successor and he said he still feels the need to prove he deserves the job.
"I feel my heart is beating quicker, more eager to get on with it and show I'm worthy of the job," he said. "It's always going to be a huge and fantastic responsibility to be manager of Manchester United.
"I probably gave up my dream when Cardiff ended as it did, and suddenly I am here.
"I still believe in sports you get what you deserve and I worked very hard. I have been myself, I love working with human beings and maybe that is what has brought me here."