Allardyce, who was appointed the Sunderland boss last week, worked with Barton at Premier League outfit Newcastle.
The 60-year-old recently told The Sun that he does not rate the 33-year-old, claiming that he ignored Kevin Nolan’s calls to sign him for West Ham United.
The former Bolton Wanderers manager also stated that Barton is not a “character” but a “liability”.
The former Queens Park Rangers midfielder has admitted on Twitter that Allardyce was right to criticise him, suggesting that he could have done better at Newcastle.
@ChronicleNUFC the disappointing thing is by the time I had got my act together, he had left as manager.— Joseph Barton (@Joey7Barton) October 15, 2015
@ChronicleNUFC too be fair to Sam, I got arrested and ended up in jail. Can't argue that at that time I wasn't a liability.— Joseph Barton (@Joey7Barton) October 15, 2015
@ChronicleNUFC huge respect for Sam as a man and a manager. Just wish I'd have played for him when I matured a while later.— Joseph Barton (@Joey7Barton) October 15, 2015
Barton started his professional career at Manchester City and moved to Newcastle in 2007. The Englishman joined QPR in 2011 and moved to Burnley on a free in the summer transfer window after leaving Rangers at the end of last season.
The 33-year-old will play an important role in Burnley’s push for promotion back to the Premier League, with the Clarets having suffered relegation to the Championship at the end of the 2014-15 campaign.