United have identified Fernandes as one of their next preferred signing after securing the services of two top targets.
The Red Devils have snatched up Daniel James and Aaron Wan-Bissaka this summer and there are also plans afoot to sign Harry Maguire and Sean Longstaff.
After an abysmal year, Solskjaer is desperate to take his former club back to the top.
The Old Trafford outfit finished sixth in the Premier League amid a multitude of problems on and off the pitch.
One of them has been in midfield, where they are looking light with the departure of Ander Herrera.
Paul Pogba and Nemanja Matic have also have their futures up in the air while the form showed by Fred last season does not fill United fans with confidence.
However, the signing of Fernandes would be a marvelous coup for the Red Devils after a sensational season for Sporting.
The 24-year-old scored 31 goals and assisted a further 18 for his side while also helping Portugal win the Nations League.
And according to A Bola, United will make a bid for the midfielder by the end of the week.
It is said should Solskjaer put further pressure on the Old Trafford board, it could help get the deal done.
United are currently on a pre-season tour down under in Australia where they have just beaten Leeds 4-0.
But with a number of transfer issues to be solved, the Norwegian coach wants his squad complete to gear them up for the new season.
However, A Bola goes on to state that Tottenham are ready to enter the race too.
Spurs have loosened the pursestrings this summer after splashing out a club-record fee on Tanguy Ndombele.
Though a fee has not yet been mooted, it is thought the initial offer could be in the area of £50m (€55m).