The former boss was at Anfield on Saturday for United's goalless draw with Liverpool on Saturday as part of his role for Dutch TV.
The Belgium worldwide has a league-high seven goals this campaign, but questions remain over his ability to produce against the big sides.
If United beat Huddersfield and Liverpool lose against Tottenham at Wembley next weekend, Mourinho will move 10 points clear of Klopp in a personal duel that will have nearly won nine games into the season.
But Lukaku, 24, dismissed suggestions he was a flat-track bully, saying it was different, and more hard, for him at former club Everton. And I go back to when [Peter Schmeichel] was playing - you had possibly six big games.
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"A lot of strikers in the league miss bigger chances than me". I learned my trade over there. For the meantime, perhaps it is a contract negotiation tactic, now that Mourinho is half way through his initial three-year deal with United. I accept that as a matter of fact.
"But sometimes it is hard as well when you play against the top teams and you play not to win and don't really create chances". Liverpool were the team who wanted to win the game, while Mourinho came for a draw. Lukaku's goals make him a crucial asset to the United team, but his arrival has not solved a long-standing problem at Old Trafford.
Mourinho's post match comments poking fun at Klopp and Liverpool's lack of urgency to win the match were created to shift attention away from a disappointing performance from his own side, as they failed to test an opposition defence that would have cracked if United had made the most of their first half openings and delivered better set-piece delivery when they had the opportunity.
"My opponent didn't open the door for me to win the game", the United manager said. "If we were to play this way... we could not do it at Liverpool".