The Belgium international scored his first goal of the Premier League season in a 2-0 victory against his former club West Brom on Saturday.
Much is expected of the 21-year-old after he sealed a permanant move to Goodison Park from Chelsea for a club-record fee reported to be around the £28 million mark following a successful loan spell last season.
Having also featured for a Belgium side which reached the quarter-finals of the World Cup in Brazil, Howard says the striker is intent on becoming the world's best.
"He wants to be the best footballer in the world, he really does," Howard told ESPN.
"That's what he thinks of himself. He gets his head down and he works, and we push him.
"He's electric, he's alive, he celebrates goals in training, everything counts to him. Everything matters. That's what you want to see. There are too many young kids that take it for granted. Romelu doesn't. He cost a lot of money but he is not arrogant, he is hungry."
Lukaku will attempt to add to his goal tally when Everton face Wolfsburg in the UEFA Europa League on Thursday.
