The Daily Star is reporting that the Norwegian manager is keen for the Red Devils to conclude a deal to land the Newcastle United midfielder this week as Manchester United continue their summer spending.
Manchester United completed their second signing of the summer over the weekend when they sealed a deal to land Aaron Wan-Bissaka from Crystal Palace.
The Red Devils are now being linked with a move to sign Longstaff, 21, as Solskjaer looks to add to his options at Old Trafford.
According to the same article, Solskjaer is “desperate” to make Longstaff his third signing of the summer transfer window for Manchester United.
And the same story says that the Red Devils are hoping to get a deal done before the squad leave for their pre-season tour of Australia and the Far East on 8 July.
Manchester United are looking to bounce back from a disappointing campaign last term in which they finished in sixth place in the table and without a trophy.
They have also signed midfielder Daniel James from Swansea City so far this summer.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news