The Reds were heavily linked with a loan move for the versatile attacker during January, but instead opted for Daniel Sturridge and Philippe Coutinho.
However, it has been speculated that they may return for the out-of-favour Paris Saint-Germain man again this summer, but they will have to fight of Sevilla to secure his signature.
The Spaniards are preparing for life without current attacking star Alvaro Negredo, who is believed to be a target for Manchester City.
If he does move on the La Liga side may look to Gameiro, who is said to be available for a knockdown fee.
PSG have deemed the pint-sized goal-getter as surplus to requirements as they step up their pursuit of Edinson Cavani.
Sevilla are currently in talks with the mega-rich Parisians, and are confident that a deal can be agreed for the player who started only seven Ligue 1 games last season.
Tottenham and Newcastle have also been linked with Gameiro, who was seen as one of France’s top emerging talents prior to his move to PSG in 2011.
Liverpool are believed to be in the market for attacking reinforcements, with Ajax’s Christian Eriksen one of their priorities.
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