The Belgian is expected to seal a move to Italy this summer after falling out of favour under Ole Gunnar Solskjaer.
Lukaku was largely used as a substitute following the Norwegian’s appointment in December, with Marcus Rashford preferred through the middle.
Antonio Conte is keen to bring him to the San Siro but Inter have so far failed to meet United’s asking price.
The Serie A giants had been keen to bring Lukaku in on an initial loan, with a structured payment deal to sign him permanently after two years.
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However, Manchester United want to recoup most of the £75m they paid for him in order to fund their summer transfer business.
According to Italian journalist Nicolo Schira, Inter are preparing a fresh bid for the 26-year-old.
He tweeted: “In the next days Inter will raise up the bid for Romelu Lukaku.
“€60M plus bonuses to Manchester United, but Red Devils want €80M at least.
“In the next days Inter will raise up the bid for Romelu Lukaku.”
Italian journalist Nicolo Schira
“The agent Federico Pastorello is working to find an agreement.
“For striker is ready a contract until 2024 by €7,5M plus bonuses a year.”
Inter’s new bid will be worth just £54m, while United want at least £72m.
Inter are prepared to pay Lukaku more than £130,000-a-week if they can strike an agreement with United.
And although the Red Devils would prefer the full fee, they may be tempted to lower their demands as the transfer deadline edges closer.
United have just three weeks to sign players ahead of the new season and it’s believed Solskjaer wants at least three new arrivals.
A centre-back is a priority and a deal for Harry Maguire looks to be progressing.
Two central midfielders are also in their sights, with Bruno Fernandes and Sean Longstaff the most likely targets.