England international Sancho has long been Solskjaer’s top target but United have been left frustrated in their attempts to sign the wide-man.
The 20-year-old is valued at £108million by Dortmund which would make him the Red Devils’ most expensive recruit ever.
United believe that asking price to be unreasonable in a transfer market heavily impacted by the coronavirus market, with very few clubs spending as freely as usual.
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And Solskjaer insists that despite’s United wealth, they will not spending “millions and millions” this summer as rivals Chelsea look set to take their outlay to around £250m with the signing of Rennes goalkeeper Edouard Mendy.
United have made one addition to their first-team squad this summer by bringing in attack-minded midfielder Donny van de Beek from Ajax at a cost of around £40m.
But they have made progress in their hopes of signing Sancho, who is thought to have agreed personal terms over a £250,000-a-week contract at Old Trafford.
Agent fees have also been agreed should a deal be brokered with Dortmund, but the German giants insist they want the full £108m up front.
United have offered to pay a guaranteed £90m over the next three seasons with another £18m in performance-contingent add-ons.
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But ex-Manchester City academy prospect Sancho will not go on strike to force a return to England, leaving it on United to submit an offer that Dortmund deem acceptable.
And Solskjaer insists United remain active in the transfer window ahead of the October 5 deadline but amid links to Real Madrid left-back Sergio Reguilon and RB Leipzig centre-half Dayot Upamecano among others, warned that United will not welcome a raft of fresh faces into the club over the coming weeks.
“We are always looking to see if there is any way possible that we can improve this squad and bring players in that will have the right level, quality, personality and fits all the criteria,” he told BBC 5 Live Sport’s Premier League Preview show.
“This summer and year has been a strange one and it is very important everyone understands this has an affect on football. Not everyone can spend millions and millions.
“I never set myself a target, like scoring 20 goals as a player. You just have to go day by day, making sure you develop and improve.
“We have a vision of where we want to get to but know so many things can come in the way. I believe in seeing how the team and club develops.
“We are a developing team, a year or so into our rebuild. We had to make some decisions last summer and throughout the season we have seen how this team is getting better and getting used to each other.
“It is never easy to have too many new players in the team, it takes time for players to gel into a unit.
“Now we have signed Donny van de Beek, he is the type of player I felt we needed in the squad and the type of person that will fit the culture and the team and will improve us for next season.
“He is a versatile player, he is clever and has plenty of energy and enthusiasm. He scores goals and that is what we need to add from midfield.
“We need goals throughout the squad, we did not score enough to challenge at the top.”
United have another week of preparation before they kickstart their 2020-21 Premier League campaign, hosting Crystal Palace at Old Trafford on Saturday 19 September (5.30pm kick-off, live on Sky Sports).
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