The 25-year-old Barton was released from a 74-day jail sentence last month after being convicted of attacking two men in December, including Jordan Spencer, who was left with broken teeth after being punched by the football player.
Spencer, now 17, was awarded $5 000 in damages by Liverpool Crown Court, but his legal team on Wednesday lodged a claim for a higher amount from Barton.
Barton is also facing civil action by a former Manchester City trainee after allegedly stubbing out a cigar in Jason Tandy's eye at the club's Christmas party in 2004.
Tandy was released by Man City in 2006 after breaking his leg during a loan spell in Denmark, while Barton left the club in June 2007.
Barton has also been charged by the FA with violent conduct for an assault on former City teammate Ousmane Dabo. He received a suspended four-month prison sentence for the assault, which took place at City's training ground in 2007.
Newcastle have resisted calls to fire Barton.