United have made recruiting the England international their top priority this summer and are the rumoured frontrunners to land his signature.
ESPN claims that Red Devils executive vice-chairman Ed Woodward is only willing to offer £50m due to the financial implications of the coronavirus pandemic.
Solskjaer is said to be worried about Woodward's negotiating tactics and wants his side to make a more realistic opening offer or risk losing Sancho to a rival club.
United used the same approach to sign Harry Maguire last summer, only to pay the full amount Leicester City had asked for two months earlier.
However, the same report indicates that, while Solskjaer is concerned, there is a real belief behind the scenes at Old Trafford that they will land their man.