Manchester United boss Solskjaer is eager for Woodward to sign Fernandes this transfer window as he plots a path back to the summit of the English game.
The Red Devils have already signed one player in Daniel James, who touched down from Championship side Swansea earlier in the month.
Manchester United are understood to be on the cusp on landing another top target, Crystal Palace star Aaron Wan-Bissaka.
Wan-Bissaka could even undergo a medical at the club’s Carrington Training Complex later in the day after an initial fee of £45million was agreed between the two teams.
The 21-year-old will not be the last player through the United door, though, with Solskjaer and Woodward eager for more.
And Portuguese newspaper A Bola claim that Woodward is under enormous pressure from his manager to land Fernandes.
The Portugal international struck a remarkable 31 goals for Sporting from the heart of their midfield last season.
And he was also key in his country’s Nations League triumph, impressing in wins over Switzerland and Holland.
Tottenham have also been pushing to sign him in the event that Christian Eriksen leaves.
But A Bola claim Solskjaer is determined to win the transfer battle and make the 24-year-old his midfield lynchpin next season.
Fernandes is said to be more inclined to play for United, rather than Tottenham, because he would be guaranteed more minutes.
The Red Devils’ need is arguably greater given the recent departures of both Ander Herrera and Marouane Fellaini.
Herrera is off to PSG after five years of service at Old Trafford.
And United sold Fellaini to the Chinese Super League back in January following the exit of Jose Mourinho, who was a big admirer of the Belgium international.
Any deal for Fernandes could be done by the end of the week.
And Tottenham moving closer to the signing of Lyon bulldozer Tanguy Ndombele could end up working in United’s favour.
Fernandes recently opened up on his future by saying: “I know nothing, I will take the next few days to understand what is true.
“We know that newspapers sometimes have to make a show to move the market and sell.
“I’m calm, I’m at a big club. I don’t need to move, I’m not trying to leave.
“If the opportunity comes to go to a team that you can’t refuse, I’ll talk to Sporting.”