The 31-year-old French ace’s contract expires at the end of the season, which has left him free to discuss a Bosman move with other interested clubs.
A switch to Inter Milan, PSG or Monaco has been heavily mooted, but now it is understood that the Reds have made a lucrative approach for the experienced defender.
Brendan Rodgers is known to be looking to strengthen his squad, and is said to be attracted to the attack-minded full-back’s experience and know-how.
He could be used as a direct replacement for the out of form Glen Johnson, or his addition may allow the England star to move to the left side of the defence with on-loan Valencia man Aly Cissokho unlikely to be retained for next season.
Sagna is believed to be looking for one last big contract and has his eye on a three-year deal.
Arsenal are said to be willing to offer him an extension, but it remains unclear as to whether or not they are prepared to meet his demands.
Chelsea are also rumoured to be watching the situation.