Manchester United boss Solskjaer has been tipped to let Lukaku leave this summer, with the Belgium international wanted by Antonio Conte.
Conte is primed to take the Inter Milan job, having been out of work since leaving Chelsea 10 months ago.
When that happens the Italian wants to lure Lukaku away from Manchester United, with Conte previously wanting the 26-year-old during his time at Stamford Bridge.
ESPN claim, though, that United will not let the Belgium international go on the cheap.
And it is said that Solskjaer would like to keep hold of the striker.
Upon succeeding Jose Mourinho in December, the 46-year-old dropped Lukaku and instead chose to make Marcus Rashford his No 9.
That decision paid off at first and Lukaku was restricted to cameo appearances as the season dragged on.
The Belgian supposedly feels concerned that Solskjaer will not make him his top striker for next season.
That is why a move to Italy appears but it seems that, despite that, Solskjaer would prefer to keep the former Everton star.
ESPN say United will also not let the player go on the cheap.
They paid an initial £75million to prise the forward away from Everton two years ago and that fee has since risen to £90m.
As a result, Inter will have to break the bank to land the centre-forward this summer.
Meanwhile, speaking at the end of last month, Lukaku fuelled talk he could be on the move away from Old Trafford.
“I don’t know, I am not here to entertain rumours,” he said.
“I am still under contract.”
Lukaku has also talked up a move to Italy, admitting to play in Serie A would be a ‘dream’.
But whether United let him go remains to be seen.