Barton claimed that an agent working on behalf of Lawwell approached him for signing talks.
The Celtic chief executive not only disputes that version of events, but says he was approached about the possibility of signing Barton.
"I think Joey's been had by a bit of a matchmaker here," Lawwell said.
"I remember it well, we had just announced Brendan (Rodgers) as our new manager and I was with our company secretary in London at the time.
"An agent called me saying that Joey was going to sign for Rangers but he would really prefer to come to Celtic and were we interested in signing him, but it wasn't something we wanted to pursue.
"These things happen to players sometimes in football, but needless to say we wish Joey well at his new club."
Lawwell's version of events differs from the extract from Barton's autobiography published in the Daily Mail.
It said that days before the player signed a two-year deal at Ibrox he "took a call from an agent, who had been contacted by Peter Lawwell, Celtic's chief executive", adding: "The message he conveyed was straight and to the point.
"'Is there anything we can do together? Can we have a conversation?'"
Peter Lawwell has dismissed claims from Joey Barton that Celtic made moves to sign the midfielder before he joined Rangers.