The 24-year-old, who joined Everton permanently in 2014, scored 25 goals and provided six assists in 37 league games last season to finish as the league's second highest scorer. His exploits helped Everton secure a seventh-placed league finish.
The Belgium international rejected a new contract at Everton in March and has since been linked with former club Chelsea and German champions Bayern Munich.
Lukaku said that he had decided on his future in June and now looks set to reunite with United manager Jose Mourinho, under whom he played at Chelsea.
Media reports also suggest that Lukaku's deal will have no impact on United forward Wayne Rooney's switch to Everton.