Reports in the Belgium international's homeland suggest a five-year contract has been agreed and although Everton have not commented, Press Association Sport understands a deal could be progressed very quickly.
The 21-year-old spent last season on loan at Goodison Park from Chelsea and was the Toffees' leading scorer with 16 goals in 33 appearances, helping the club finish fifth.
Martinez made Lukaku one of his primary summer targets but the player's involvement in the World Cup and extensive negotiations with Chelsea have taken time.
The striker tweeted a picture of himself on a plane with the message "Time to write a new chapter.....", although whether the inclusion of a partially-obscured exit sign behind the player's head was intentional or accidental is open to debate.
Lukaku's Belgian team-mate Marouane Fellaini, who coincidentally is currently Everton's club-record signing after a £15million fee was paid to Standard Liege back in 2008, added further support to reports.
The Manchester United midfielder's post on Instagram said "CONGRATULATIONS to my man @romelu_lukaku17 on signing for @officialeverton !! Big future ahead" and also featured Lukaku's plane picture alongside an Everton badge and a shot of the pair in battle against each other in the Premier League last season.
Source : PA
Source: PA