The European champions, according to The Sun, have halted contract negotiations with Victor Valdes, their first-choice goalkeeper, after he demanded £150,000 a week.
However, Pep Guardiola, the Barcelona manager, has refused to give in to Valdes and now wants the former Barca youth player to make a return to the Nou Camp.
The move will anger Rafael Benitez, the Liverpool manager, who has already had to fight off approaches from the Spanish champions and Real Madrid for midfielders Javier Mascherano and Xabi Alonso.
Reina, whose father played for Barcelona and still lives in the Catalan city, joined Liverpool in the summer of 2005 for £6m to replace Jerzy Dudek and has proved popular with the fans.
Meanwhile, Andrea Dossena, Liverpool's Italian left-back, has said he wants to stay at the club after he was linked with a move to Juventus and believes Benitez is close to bringing the league title back to Anfield after a 20-year wait.
Dossena, who joined the club from Udinese last summer and has struggled since his move to England and started just 12 games in the Premier League last season, said: "I learned what the boss wants from me and I hope next season is better than last season.
"I never said I wanted to go away from Liverpool. I'm happy in Liverpool and I want to stay."
Dossena, who scored Liverpool's fourth in their 4-1 thrashing of Manchester United at Old Trafford last season, added: "There's not a lot of difference between us and them [United].
"Now Manchester United have lost Cristiano Ronaldo maybe that's better for us."