Just two years after arriving from Everton in an initial £75million deal, the 26-year-old is understood to be welcoming interest from the Serie A giants.
Antonio Conte's side met United in recent days for discussions, but PA understands that Inter have been struggling to fund a deal for the Belgium international.
The striker missed Saturday's 2-0 friendly win against Perth Glory due to a knock and fuelled speculation about his future as he left Optus Stadium, telling reporters: "You guys will know something next week".
Lukaku's brief comment came as Solskjaer faced questions about the frontman's fitness and happiness at the post-match press conference.
"There's been so many speculations but, yes, he should be fit for Wednesday (against Leeds)," the United boss said.
"He trained with the team yesterday, but it wasn't useful today to risk him, to be fair."
Quizzed on whether Lukaku will still be a player at the start of the season, Solskjaer added: "Well, let's see when the season starts. We've not had any bids that we're considering.
"All my conversations with the players I will keep confidential. That's between us.
"That's my privilege and I can promise them that I won't tell (about) our discussions."
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