British media reported that the fee for the 24-year-old is about 75 million pounds, making him the Old Trafford club's second most expensive signing after last year's world-record deal for France midfielder Paul Pogba.
"Romelu is a natural fit for Manchester United. He is a big personality and a big player," United manager Jose Mourinho said in a statement.
"It is only natural that he wants to develop his career at the biggest club. He will be a great addition to the group and I know they will make him very welcome. I am really looking forward to working with him again."
Lukaku, who also attracted interest from his former club Chelsea, scored 25 league goals in 37 matches for Everton in the last campaign, four behind Tottenham Hotspur's Golden Boot-winner Harry Kane.
"I would like to start by thanking Everton and the fans for the last four wonderful seasons, I have made some special friends and we have shared some amazing moments," Lukaku said.
"However, when Manchester United and Jose Mourinho come knocking at the door it is an opportunity of a lifetime and one that I could not turn down.
"I cannot wait to run out at Old Trafford in front of 75,000 fans but before that pre-season is where the hard work starts and I am looking forward to that first training session."
Lukaku, who has scored 20 goals for Belgium, rejected a new contract at Everton in March and will reunite with Mourinho who managed him at Chelsea.
(Writing by Neil Robinson in London, editing by Ed Osmond)
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