The former Manchester City and Newcastle United midfielder is currently looking for a new club, having been released by Queens Park Rangers after their relegation from the Premier League.
United have been linked with the controversial playmaker for the last couple of weeks but the speculation is increasing that Leeds could be close to getting a deal over the line.
Barton would become Leeds' first signing of the summer under manager Uwe Rosler if a deal does go through, while the Whites are also thought to be chasing Motherwell striker Lee Erwin.