Managing director Ian Ayre flew to Barcelona on Wednesday to speak to the Uruguay international's agent Pere Guardiola.
However, any potential deal is some way off yet as it seems unlikely that the 26-year-old will agree to extend his stay until he knows whether the club have qualified for next season's Champions League.
Suarez has been in scintillating form for Liverpool this season and is the Premier League's leading scorer with 17 goals.
The 26-year-old looked set to leave Anfield last summer, with Arsenal lodging a bid of £40,000,001 in the mistaken belief it would trigger a release clause.
A move did not materialise and Suarez, whose current contract expires in 2016, seems fully focused on restoring Liverpool's place among Europe's elite.
Former Reds midfielder Dietmar Hamann believes the club need a long-term personal commitment in addition to any new contract.
"What must not happen is we have a situation next summer as we had last summer," said Hamann.
"He could sign a new contract now, take the £200,000 a week, the team finishes outside the top four and you have the same shenanigans.
"I want him to stay but there needs to be clarity. He has got to understand what it means to commit.
"If there is a clause that he can go for £60million, £80million and someone pays it, that is obviously fair enough, but you can't have a situation again like you had last summer."
Suarez has scored 11 in his past six appearances but Hamann questioned what the striker's mood will be like should things take a turn for the worse later in the season.
"The way he is playing at the moment you have got think he is very happy but you have got to be careful because the team is on a good run," he said.
"It was probably the most impressive performance from any team this season on Sunday (beating Tottenham 5-0).
"They have been absolutely fantastic and in this form you have to think they will finish in the top four, but things can change."
Liverpool defender Mamadou Sakho has praised Suarez's attitude and insisted he is a great team player.
"For sure, it's tough against Luis because he's a world-class striker. That helps us improve," Sakho told the club's official website.
"I'm lucky enough to play with him, but the most important thing is that Luis gives everything for the team.
"He really makes himself available for the group just like all the players here and that's why Liverpool are continuing to make progress with every game."