The Millers manager has had little success with his two previous bids for the former Field Mill trainee but Moore seems more hopeful a deal can be reached with The Stags next week.
"I'm still hopeful that something will happen next week," said Moore.
"He's a player who I like a lot and I'd still like to have him down here.
"We need to have further words with his agent (Colin Gibson) but we're still hanging on in there."
The Division Three side have rejected bids of £150,000 and £175,000 for Lawrence and it remains to be seen if Moore's third offer will change their minds.
Earlier this week, Stags chairman Keith Haslam admitted his displeasure at Gibson for having rekindled Rotherham's interest in the player.
"To the more cynical, a football agent's job is to move players around - this is how they earn their money," said Haslam.
"Realising he had a gem in Liam, Colin Gibson contacted Rotherham which stirred the situation up again and prompted Ronnie Moore's fresh offer."
Meanwhile, Mansfield's young striker Chris Beardsley has joined non-league side Worksop on loan while fellow hit-man Craig Mitchell has gone to Conference club Northwich Victoria on loan.
Alex Baptiste has returned to Field Mill following a loan spell at Tamworth.
