The Daily Record says Lawrence expects to complete his move to Parkhead when Aiden McGeady finally signs for Spartak Moscow this week.
"Well, that's what I've been told," he said. "I want a new challenge and, if it comes about, I will be chomping at the bit to achieve things at Celtic because they need to push this year and win things again.
"I have had some little bits of banter with Aiden as well.
"I spoke to him the other day and fired him a text because I knew he was off to Russia. I'm not telling you what I said though.
"It's a massive, massive move for him and it's an important decision both football-wise and for him and his missus."