The 32-year-old midfielder has been looking for a club since his release from QPR at the end of last season.
Burnley boss Sean Dyche held preliminary talks with Barton in pre-season, but the Clarets are now to step up their interest after the midfielder saw a move to West Ham break down earlier this month.
Barton has been linked to Rotherham United, but Millers boss Steve Evans scotched talk of that move, saying “he’s a fantastic footballer – but I don’t think we can afford Joey’s petrol, never mind Joey.”
Nottingham Forest was another possible destination for Barton but Reds boss Dougie Freedman told him the club could not afford his wage demands.
However, the Liverpool-born former Manchester City and Newcastle man could still be poised for a return north with Burnley, who have five points from their opening four Championship games.