The Sun says Kop director of football Damien Comolli tried to land the 21-year-old Algeria international in January but was knocked back by his club.
Now Anfield officials have re-opened negotiations with Sochaux and the player's advisors and are set to fly to France.
Boudebouz is rated at £5million and scored eight goals in 38 league games last season to help clinch a Europa League place.
He said: "Things happen quickly in football. I will only say that I am no longer certain of staying put at Sochaux."