The Sun, the Daily Mirror and the Daily Mail all report a possible £12m deal when the January transfer window opens.
Manager Rafa Benitez has said he is looking to spend in the New Year, saying: "We are working hard to find the right players in January."
Villa, who has 53 goals in 85 caps, formed a devastating partnership with Torres in Spain's Euro 2008 and 2010 World Cup triumphs, though he missed Euro 2012 with a broken leg.
The 31-year-old has made only a handful of starts for Barcelona since returning from injury this season and the Liga giants could be ready to part with the striker, who cost them £33m in 2010.
Torres has regained his scoring form since Benitez replaced Roberto Di Matteo as Chelsea manager, finding the net seven times in his last six games.
However Benitez believes Torres can deliver even more, and wants more support for his only recognised senior striker.
Despite Villa's increasingly marginal role - thanks in part to the record-breaking goalscoring feats of Lionel Messi - Barcelona's sporting director Andoni Zubizarreta has said: "In January, we will keep Villa."
Meanwhile, The Sun claims Chelsea have told Frank Lampard to find a new club.
The 34-year-old, who moved to Stamford Bridge in 2001, is out of contract in the summer and Chelsea reportedly have no intention of offering him a new deal.
The Sun quotes a source as saying: "Frank is absolutely distraught. He wants to keep playing for Chelsea but they've told him to find another club in the January transfer window. He can't believe it."
Lampard became Chelsea's all-time leading top-flight goalscorer in Saturday's 8-0 mauling of Aston Villa, which was his 500th Premier League start.
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