This has led to speculation that Brendan Rodgers may allow him to leave, after appearing to be unconvinced by the Spaniard, who has made a shaky start to the current season.
The Northern Irishman may look to Ochoa, who will cost around £5m, as a replacement.
The Mexican has become a key figure at Ajaccio, helping the club maintain top-flight status last season after winning promotion to Ligue 1 in 2011.
Liverpool have also been linked with Birmingham’s Jack Butland, but are believed to be unwilling to match the Championship club’s asking price for the 19-year-old.
Ochoa would certainly be a more affordable option and would bring greater experience, having represented Mexico at senior international level on 50 occasions.
It is unlikely that the Reds will make a January move, with Reina’s immediate future appearing to be with Liverpool.
The 27-year-old started his career with Club America in his homeland, making over 200 first-team appearances.
He swapped Mexico for France in 2011, signing with Ajaccio.