Barça's current number one Victor Valdes announced this week he will leave the club when his contract expires in 2014.
The Catalans want to find a replacement this summer, and the Mail on Sunday claims they want Reina, who started his career at the Camp Nou.
Reina played 49 times for Barcelona before joining Villarreal in 2002, and subsequently moving to Anfield in 2005.
Although Reina's form has dipped at times this season, he is a popular (though largely non-playing) member of the Spanish national squad.
His acquisition would continue Barça's habit of re-signing players trained at the club, after they brought Gerard Pique and Cesc Fabregas back.
Reina signed a six-year contract with Liverpool in 2010.
Former Barcelona manager Pep Guardiola has identified his main transfer targets for Bayern Munich, the Sunday Mirror claims.
Guardiola reportedly wants Luis Suarez and Radamel Falcao - which makes sense, since both are excellent at football.
Frank Lampard faces a dilemma over whether to join AC Milan or Los Angeles Galaxy as his Chelsea contract winds down.
The midfielder has not been offered a new contract at Stamford Bridge, and looks set to leave after 11 seasons there.
The 34-year-old has been in talks with David Beckham's former team, LA Galaxy, and is thought to favour a change of scenery to California.
However, such a move would almost certainly end his England career.
Milan, however, would enable Lampard to continue playing at the highest level in Serie A and European competitions.
The Sunday papers seem split on Lampard's destination - the Sunday Express have him going to LA, while The People and the Mail on Sunday rate Milan as more likely.
Paper Round staple Wilfried Zaha could be appearing here for a while longer.
The Crystal Palace winger was thought to be close to agreeing a move to Manchester United, but the Sunday Mirror reports that the two clubs have still not agreed a fee.
United's offer of £10m is approximately half what Palace want for the coveted 20-year-old.
Another player constantly on the verge of a move somewhere is Mario Balotelli - the Observer says his move to AC Milan has stalled due to Manchester City's unrealistic £31 million asking price.