The centre-back made a sliding challenge on the striker in the second half of the stalemate at Anfield, after which Lukaku seemed to struggle to extricate himself from the tangle of legs.
Lovren rolled away with his hands to his face as Lukaku freed himself and replays of the incident suggested the Belgium international caught him with the heel of his boot.
The officials took no action at the time and it is reported the Football Association will not pursue a case against Lukaku, but Lovren believes he intentionally made contact.
Speaking ahead of Liverpool's Champions League clash with Maribor, he said: "I made a tackle and I felt he was over me and could have moved away. My honest view is that he did it on purpose.
"It is not my decision. He seemed nervous during the game and maybe that's why. Normally if you do it, you apologise.
"It happened and it's over. Nobody can change it."
The draw meant Liverpool have only won one of their last eight games in all competitions, including the Champions League, in which they drew their opening group games against Sevilla and Spartak Moscow.
Lovren says frustrations are still high that they could not get all three points against Jose Mourinho's side but he insists they should take confidence from the way they played.
"There is still frustration because we played much better than United," he said. "We had chances that we didn't take but after each game you have to take the positives.
"We showed to everyone that we can play against big teams. There is no bad mood within the dressing room. We are confident."