The Anfield boss' priority is to sign an imposing centre forward to take the pressure off Fernando Torres in January and has considered the likes of 6ft 3in CSKA Moscow target man Tomas Necid.
But the guile of 5ft 8in Kleber has attracted them in the past and Palmeiras are now perpared to let him leave on the cheap after going out of the Copa Sudamericana.
Roy's boy: Palmeiras' striker Kleber could be offered to Liverpool on the cheap
Kleber, who has also previously been linked with Everton, claimed last year that Liverpool had made a £6million bid for his services while still on Cruzeiro's books.
Palmeiras are willing to do business after a disappointing year to save cash on one of their highest paid players, even though he is their highest scorer in Brazil's Serie A with 10 goals this season.
The 27-year-old is on the Sao Paulo club's books till 2014 but claims he is owed three months' salary, which could pave the way for a free transfer.
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After talks with Palmeiras director of football Wlademir Pescarmona, Kleber said: 'He (Pescarmona) told me that if I want to leave the club, I can.'
Kleber is uncapped but does not need a work permit as he is understood to have an Italian passport.
He has previously had a successful four-year spell in Europe, averaging a goal every other game with Dynamo Kiev and winning the domestic league twice and cup three times before returning to Brazil in February last year.
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