United have identified Fernandes as one of their next preferred signings 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 and will be plying their trade in the Europa League next season.
But that has not stopped executive vice-chairman Ed Woodward from splashing the cash, with three more signings mooted.
A deal for Fernandes would be a marvellous 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 while the club are pursuing a move, the Primeira Liga outfit have left United angry and frustrated.
The Evening Standard report that the Red Devils believe that they are being used to drum up a bidding war on the midfielder.
Sporting are resigned to letting their prized asset go this summer.
But their asking price has inflated from £58m to £70m in recent months and United believe that their name is being tagged to attract more suitors for the Portuguese ace.
Fernandes was in the headlines on Wednesday night after scoring a potshot from distance against Liverpool in their 2-2 draw in New York.
And Sporting manager Marcel Keizer hinted that his main man could be heading for the exit door.
“Everybody would like him to stay,” he said.
“The situation is that he is here, but we don’t know what the future will bring.
“We know he plays very well, we know that clubs like him, so we have to see what happens.
“The job of the coach is to realise that some players will go before the end of the window. Every player can go, and we can also buy players.”