Inter Milan sporting director Piero Ausilio held talks with Ed Woodward in London on Thursday to discuss a swoop for the Belgium international.
The Serie A club want a loan deal, paying £9m and then a further £54m in two instalments, but Manchester United are demanding a straight-cash deal worth £90m.
Lukaku is in Australia for the club’s pre-season tour but missed the 2-0 win against Perth Glory due to a minor injury.
Questioned about his future as he left the Optus Stadium, Lukaku said: “You guys will know something next week.”
Lukaku fell behind Marcus Rashford in the pecking order after Ole Gunnar Solskjaer succeeded Jose Mourinho.
He sparked speculation of a move to Inter Milan when he described Antonio Conte as the “best coach in the world”.
But the San Siro outfit are struggling to raise the funds needed to complete the deal and Solskjaer has suggested an exit is not close.
“There’s been so much speculation but he should be fit for Wednesday’s game [against Leeds],” the United boss said.
“He trained with the team yesterday but it wasn’t useful today to risk him, to be fair.
“Let’s see when the season starts – we’ve not had any bids we’re considering.
“All my conversations I will keep confidential, that’s my privilege and I can promise them that I won’t tell them our discussions.”
Inter Milan could raise funds by selling Mauro Icardi, who is interesting Juventus.
The Argentina striker could be on the move after leaving the team’s pre-season training camp to return to Milan.
A statement from the club read: “Mauro Icardi will today return to Milan from our training camp in Lugano.
“The club and the striker have come to this decision by mutual agreement.
“Icardi will continue his pre-season training over the next few days and will take no part in our Summer Tour in Asia.”