"Lukaku is ours" chanted Inter Milan ecstatic fans who turned out at the airport to welcome him on his arrival during the night.
The 26-year-old has not played a single minute of United's pre-season campaign as talk of a transfer to the Serie A intensified, with Inter Milan the hot favourites.
Italian newspapers have said the fee could be 65 million euros ($73 million), plus bonuses of around 10 to 13 million euros.
Sports newspaper Gazzetta Dello Sport said Lukaku travelled to Milan with his agent and friend Federico Pastorello and was to undergo stringent medical tests on Thursday.
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Inter coach Antonio Conte said on Sunday he was "very hopeful" of finalising a deal to bring the former Everton star to the San Siro.
"Lukaku is a good player, a striker who I had tried to buy when I was coach at Chelsea" from 2016 to 2018, Conte said.
According to Gazzetta Dello Sport, Manchester United initially refused a deal and frenzied contacts between the two sides have taken place in recent days.
The player was in Brussels on Monday and Tuesday training with his boyhood club Anderlecht. Local papers showed pictures of him in the Belgian club's colours.
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Inter have been in the market for a centre forward since the start of the transfer window.
But the club has been competing for his services with fellow Serie A side Juventus, who Italian media said had offered Manchester United an exchange with Paulo Dybala.
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