Manchester United boss Solskjaer could lose Lukaku to Inter Milan, with the Serie A giants keen to lure the Belgium international to the San Siro.
The former Everton star has been told he will play second fiddle to Marcus Rashford in attack next season and, consequently, is keen to make the move to Italy.
Manchester United have so far held firm, though, with the Red Devils holding out for £90million in order to let the striker go.
Inter are financially strong but even they have doubts as to whether or not Lukaku is worth a punt for that price.
And the Times say that, if Lukaku goes, Solskjaer is adamant that Woodward must fork out on a replacement.
United, when it comes to strikers, have several player capable of operating in a No 9 role.
Rashford can play there, while Solskjaer has trialled using Anthony Martial in a central position.
Alexis Sanchez, who is expected to stay at the club amid United’s failure to find a buyer, had the best season of his career when playing as a striker for Arsenal.
And teenage prodigy Mason Greenwood has made a name for himself at academy level operating at the tip of the attack.
Solskjaer, though, would want another signing.
But whether Woodward, who is trying to push through deals for Harry Maguire, Sean Longstaff and Bruno Fernandes, can force another remains to be seen.
Lukaku has blown hot and cold during his time at United.
He hit 27 goals in his first campaign at the club as they finished second in the Premier League and reached the FA Cup final under Jose Mourinho.
But he struggled last term and could now leave.
Inter boss Antonio Conte recently opened up on the striker, saying: “You know very well I like this player [Lukaku].
“Today Lukaku is a player of United and this is the reality. This is the truth.
“For sure it’s not easy to play without a striker at the beginning of the season like this.
“The market is still open.”