Luis Suarez will be allowed to hold talks with Arsenal but Liverpool have no intention of selling their striker until his £50m-plus valuation is met.
Arsenal's club-record bid of £40m plus £1 was firmly rejected by Liverpool but Suarez now wants to hold talks with the north London club.
The approach triggered a clause in his contract that means he has been told of the bid and can now talk to Arsenal.
But Liverpool will not sell him until Arsenal up their offer significantly.
On Wednesday Suarez made his first appearance for Liverpool since his infamous bite on Chelsea defender Branislav Ivanovic's arm last season, coming off the bench in an 18-minute cameo during the Premier League club's 2-0 win over Melbourne Victory at the MCG.
With Liverpool 1-0 up, the Uruguayan laid on the Reds second for new signing Iago Aspas late in the match.
After the game Liverpool manager Brendan Rodgers said: "There's nothing new to report, he [Suarez] is very much a Liverpool player and over the course of the next couple of weeks we've got to get him up to speed."
Rodgers did, however, remind Suarez of the debt he feels the player owes the Liverpool fans who have stood by him after two seasons of controversy.
"The support he has received from the supporters and the people of the city of Liverpool has been unrivalled," Rodgers added.
"In this period of time he's missed a lot of games for the club through various reasons. The people have stood by him like a son and really looked after him. Whatever happens in the coming weeks that will be in his mind because it's something you can never forget."
Liverpool have now turned down two offers from the Gunners, who are keen to boost their attacking options, while Real Madrid, now managed by Carlo Ancelotti, also remain interested in the player but have yet to make a formal offer.
Following Arsenal's most recent approach, a post on Liverpool owner John Henry's Twitter page read: "What do you think they're smoking over there at Emirates?"
It is unclear whether Henry is talking about Arsenal's attempts to sign Suarez or their valuation of the player.
While Liverpool remain determined to keep Suarez, who scored 30 goals for the club in 44 appearances last season, their resolve is likely to be tested if the bids continue to rise. Should Arsenal persist and have an improved bid accepted, it would more than double their previous highest initial outlay on a transfer.
The Gunners, whose first offer for Suarez was £30m, paid £17.5m to Sevilla for Jose Antonio Reyes in 2004.
Suarez wants to leave Anfield to play for a Champions League club despite only signing a new long-term deal last summer.
Speculation has grown about Suarez's future since he was punished with a 10-game ban at the end of April for biting Ivanovic.
The striker still has to serve six games of that suspension and was also banned in 2011 after being found guilty of racially abusing Manchester United's Patrice Evra.
Suarez joined Liverpool from Ajax in January 2011 in a £22.7m deal.