He was frozen out by United manager Ole Gunnar Solskjaer and will be welcomed at the San Siro by new Inter coach Antonio Conte, who tried to take him back to Chelsea in 2017.
England worldwide Hudson-Odoi has entered the final 12 months of his contract and, while Lampard wants the 18-year-old to stay in west London, he says there has been no progress on a potential new deal.
Conte has also brought in the likes of Valentino Lazaro, Nicolo Barella and Matteo Politano to improve his squad, while it remains unclear what Lukaku's arrival means for fellow striker Mauro Icardi.
Inter Milan are believed to have submitted a new and improved bid for Romelu Lukaku in the hope of finally signing the wantaway Manchester United frontman.
Lukaku hit the ground running in the north-west but has slowly fallen out of favour at United largely due to issues off the pitch, although goals have also dried up over the last 12 months.
"And was his style suited to playing football for Manchester United?".
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"Of course Rom has a good record and statistics, he's one of the top number nines around when you want to play with that kind of striker, a target man", the United manager said.
There was a chance that Lukaku wasn't going to be the only departure from Old Trafford yesterday.
At 32, Mandzukic would have been a decent short-term option, but one that would best work if Man United actually had a designated out-and-out striker.
Lukaku worked out at former club Anderlecht's facilities while the furor around him persisted.
Lukaku's arrival will cast further doubt over the future of striker Mauro Icardi, who was stripped of the captaincy a year ago.