Lukaku is a target for Inter Milan but United have told the Serie A side they will need to pay more than £79m to sign the 26-year-old Belgium international, increasing the prospect of him staying at Old Trafford.
Inter sporting director Piero Ausilio was in London on Thursday to meet United officials but was left in no doubt that their current proposal would not be good enough as they struggle to meet the Premier League club’s valuation.
Lukaku was left out of Saturday’s 2-0 win over Perth Glory after picking up a minor injury but Solskjaer expects him to be fit for Wednesday’s game against Leeds .
“He should be fit for Wednesday, He trained with the team yesterday but it was not useful today to risk him,” Solskjaer said.
Asked if Lukaku would still be at the club when the transfer window closes, Solskjaer replied: “Let’s see when the season starts. We have not had any bids that we are considering.”
Later, Solskjaer issued a further update, telling reporters “you guys will know something next week”.
United have made two signings so far this summer, bring in Daniel James from Swansea for £18m and Aaron Wan-Bissaka from Crystal Palace for £50m.
Their pursuit of Leicester City’s Harry Maguire remains “very much alive”, a source close to negotiations has told Sky Sports News this week.
They also remain interested in signing Sean Longstaff from Newcastle – though the two clubs are a long way apart in their valuation of the player – and Sporting Lisbon midfielder Bruno Fernandes.
Solskjaer is hopeful United can add to their squad before the transfer window shuts on August 8.
“Timing is obviously an issue but we are working as I have said before. Of course we are optimistic,” Solskjaer added.
Paul Pogba has been linked with a move to Real Madrid after saying he wanted a ‘new challenge’ last month but Solskjaer says the French midfielder is among the players he is considering naming as captain for next season. United’s former manager Jose Mourinho would certainly do the same.
Asked about the captaincy, Solskjaer said: “We have got a few possibilities there. When the season starts we will announce [someone].
“Ashley [Young] will wear it when he is playing until we decide on someone.”
Asked if Pogba is a consideration, Solskjaer said: “Yeah.”