Wolves visit Old Trafford on Saturday, before a trip to West Ham and a home fixture against Newcastle, by which time Lukaku could be the clear leader in the race for the Golden Boot.
Lukaku goes into this weekend with four Premier League goals, one behind Chelsea’s Eden Hazard at the top of the standings, but has not had an entirely smooth start.
The Belgium striker has scored in each of United’s past two league matches, with both goals in a 2-0 win at Burnley on September 2 and another in last weekend’s 2-1 victory at Watford.
Yet he continues to face accusations that he is a ‘flat-track bully’ because of his disappointing record against the rest of England’s big six teams.
Of the 20 Premier League goals Lukaku has scored for United since arriving from Everton in July 2017, only one has come against the teams who made up last season’s final top six – when Chelsea were defeated at Old Trafford in February.
Perhaps, then, a run of fixtures against opponents lower down the table will offer a chance to bolster his goal total.
He may also be helped by the fact Liverpool’s Mohamed Salah and Tottenham’s Harry Kane have not, so far, reached the high standards they set last season, while Sergio Aguero, who has three league goals to date, is likely to be rotated with Gabriel Jesus at Manchester City.
“You always compete every year against Aguero and Kane, and Salah this year. You always think about (the Golden Boot), being a striker,” Lukaku said.
“But then, while I want to score as much as I can, I want to win championships. That’s why I signed for Manchester United.”
Last season, Lukaku was the sixth-highest scorer in the Premier League with 16 goals, half the total that Salah achieved to win the Golden Boot.
There is an argument, though, that the Belgian was still adjusting to life as a United player the last term.
His all-around game has improved since his days at Everton, something that Mourinho noted after the win at Watford.
Mourinho was quick to credit Lukaku, not for his goal, but for running back half of the length of the pitch late in the game to make a sliding tackle as United held on to their lead.
According to Mourinho, that represents the spirit of the team.
United appear to be in good spirits at present, having won three successive matches for the first time since April.
The victory at Watford was followed on Wednesday by a 3-0 defeat of Swiss side Young Boys in the Champions League, and Mourinho’s recent grumbles about a lack of adequate close-season transfer market activity have quietened.
A consistent goalscoring run from Lukaku over the next fortnight will further improve both the mood and the league position; costly defeats by Brighton and Tottenham during August mean that United go into the weekend lying eighth, six points behind leaders Chelsea.
If Lukaku can keep on scoring regularly beyond that – with a league trip to pace-setters Chelsea to come later in October – then he may have a genuine chance of winning the league titles he craves.