Managing director Ian Ayre flew to Barcelona on Wednesday to speak to the Uruguay international's agents about the possibility of a new deal, Press Association Sport understands.
It seems unlikely the 26-year-old will agree to extend his stay beyond the 2016 end of his current deal until he knows whether the club have qualified for the Champions League.
Even then that may not necessarily be enough to keep him but, having previously pledged his support until the end of the season, the league's leading scorer with 17 goals seemed fully focused in restoring Liverpool's place among Europe's elite.
That is a far cry from the summer when he was very outspoken with his frustrations about not being allowed to leave for a Champions League club, especially after Arsenal lodged a bid of £40,000,001 in the mistaken belief it would trigger a release clause.
But while everything seems relatively stable with Suarez at the moment 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 the German, a Champions League winner with Liverpool in 2005.
"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 17 goals in 11 matches this season since returning from a 10-game suspension, carried over from the previous campaign, for biting Chelsea's Branislav Ivanovic.
At the weekend he won the Football Supporters' Federation's player of the year and with Liverpool second in the league, just two points behind Arsenal, confidence and optimism is high at Anfield.
However, they have a tough Christmas schedule with away trips to Manchester City and Chelsea after Saturday's visit of Cardiff and those two games could have a significant bearing on the rest of the season.
Suarez is scoring goals for fun with 11 in his past six appearances - including four against Norwich - since the defeat to the Gunners in early November.
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 told talkSPORT.
"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."
Suarez's team-mate Mamadou Sakho has seen no reason to doubt the player's commitment.
"The most important thing is that Luis gives everything for the team," he told liverpoolfc.com.
"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."
Any contract offer for Suarez would have to reflect his pivotal role in the side and would probably make him the highest-paid player at the club.
That would also be a significant statement of intent by owners Fenway Sports Group, who have built their business model more on reducing expenditure on salary - highlighted by the loaning out of previous first-choice goalkeeper Jose Reina to Napoli and offloading the likes of Andy Carroll and Stewart Downing to West Ham at a considerable loss.
Captain Steven Gerrard is currently the club's highest-paid player but with the 33-year-old midfielder currently contracted only to 2015 and the increased prospect of a money-spinning return to the Champions League there are exceptions to be made.
The Uruguayan's value has rocketed in the last four months and while that may not deter the likes of Real Madrid, who were reported to be interested before their world-record transfer for Gareth Bale, Liverpool are doing all they can to hold on to their prize asset.