Barton, 34, is currently serving a three-week club suspension following a row with boss Warburton and teammate Andy Halliday.
United States international Perry Kitchen has been named the club captain of Scottish club Hearts for this season.
On Monday, Barton was suspended for three weeks after reportedly holding talks with manager Mark Warburton and board members.
Lawwell's rebuttal came just hours after it emerged that Barton, 34, was facing a ban after allegedly breaking strict betting rules.
Rangers midfielder Joey Barton is under investigation for allegedly breaking football betting rules, Press Association Sport reports.
Rangers midfielder Joey Barton has been suspended for three weeks, the club have announced.