Suárez, who wanted a transfer in the summer, has been in scintillating form for the Reds this season, scoring an astonishing 17 goals in 12 games.
And the Reds' managing director, Ian Ayre, flew to Barcelona on Tuesday for preliminary negotiations with Suárez's agent, Pere Guardiola, according to several national newspapers.
Suárez signed a new long-term contract in August 2012 and has two-and-a-half-years remaining on his current deal.
He reneged on his desire to leave Merseyside in pre-season "because of the affection of the fans", however his capricious nature could result in a departure following next year's World Cup.
Suárez has not played in the Champions League in over three years, and although Liverpool could go top of the Premier League table at the weekend with victory over Cardiff, their main target remains a top four finish.
Brendan Rodgers' side have seized on Manchester United and Tottenham's poor form, however the season is still at a premature stage and Everton pose further competition.
Suárez was the subject of a £40,000,001 bid from Arsenal in July, and despite his supposed qualms about the English media, he was keen to join Arsène Wenger's side.
The 26-year-old initially claimed he wanted to leave the country because he no longer felt "comfortable" in the Premier League.
"My reason for leaving is not the money. It is my family and my image. I don't feel comfortable here anymore," the Uruguay striker said in a TV interview in May.
Suárez has served 19 matches worth of suspensions, for making an obscene gesture at Fulham fans, racially abusing Patrice Evra and biting Branislav Ivanović, in his three years with Liverpool without ever being sent off.