The Reds have been short of attacking options since Rodgers' sanctioned Andy Carroll's loan move to West Ham towards the end of the summer transfer window.
And the Liverpool manager has hinted he is willing to spend in January to bring in some extra firepower. One of those transfer window targets is believed to be Barcelona forward Sanchez.
The Chilean international joined the Nou Camp club from Udinese last summer, and impressed in his debut term with the Catalan giants.
But Sanchez faces stiff competition for a first-team place, and with David Villa returning from injury, Sanchez's role in the starting XI could be diminished over the coming season.
Under new Barca boss Tito Vilanova Sanchez has found himself on the bench for four of his side's first seven league games, and talkSPORT reports the Chilean could fall out of favour at the Nou Camp.
Should this be the case, Rodgers could be ready to step in with a bid to tempt Sanchez to Liverpool. The Reds have endured a difficult start to the season, and found themselves desperately short up front after Carroll's departure, a crisis exacerbated by injury to Fabio Borini last week.
The former-Roma striker fractured his foot on international duty with the Italy under-21 squad, ruling him out for up to three months, and leaving Liverpool with only one fit senior striker - Luis Suarez.