It was also reported the so-called "first daughter" "made an emphatic case for a full-throated apology", according to several witnesses who were present for the crisis discussion that occurred after.
But the Times reported Mr Trump only agreed to say sorry for any offence he had caused.
In the footage, Mr Trump talked about being "automatically attracted" to beautiful women and claimed you can "do anything" when you're a star.
The recording showed him saying: 'You know, I'm automatically attracted to beautiful; I just start kissing them. It's like a magnet. Just kiss. I don't even wait.
"And when you're a star, they let you do it. You can do anything. Grab 'em by the p---y. You can do anything.'
The soon-to-be President made an unusual videotaped apology after the story made headlines around the world.
"Anyone who knows me knows these words don't reflect who I am ... I said it, I was wrong, and I apologise," he said.