Reds managing director Ian Ayre flew to Barcelona on Monday to meet with Suarez's agent about offering the Uruguayan a new and improved deal.
Suarez is contracted to the Merseyside club until 2016 but the Reds are keen to prevent another summer of uncertainty over his future at Anfield.
It is believed Liverpool will need to offer the 26-year-old close to £200,000 per week to persuade Suarez to stay, almost double his current contract.
But given Suarez's sublime form this season and importance to the Reds on-field success, the club may be willing to come through with the cash.
"He is really happy here. He is playing with a smile and enthusiasm," Liverpool boss Brendan Rogers said.
"The whole profile of the club fits him perfectly. The supporters idolise him and you can see the connection there.
"There are not many like him. There are some outstanding talents but when you have actually got one, you have to fight tooth and nail to keep it.
"That is something we will look to do. We want him to stay and with his contract running for two-and-a-half years I am sure there will be something between now and the end of the season."
Spanish giants Real Madrid are known to be interested in luring Suarez to the Bernabeu, while Arsenal made two bids for his services in the summer and may come again.
Suarez has scored 17 goals in just 11 league games this season after he missed the first six games due to his 10-match ban in April for biting Chelsea defender Branislav Ivanovic.
