Arsenal late on Tuesday night made a second bid for the Uruguay international that amounted £40 million (Dh225.6m) plus £1 that Suarez believes has triggered a clause in his contract allowing him to speak with potential employers, according to BBC Sport.
Earlier, Liverpool owner John Henry appeared to have ridiculed Arsenal's bid in a seemingly dismissive tweet.
The "What do you think they are smoking over there at Emirates" message was written on the American's Twitter account just after 2am local time in Boston (10am in the UAE), seemingly in response to the Gunners' offer.
While it may seem out of character – club officials have not yet been unable to verify it was sent by Henry himself because of the time difference to the US – it undoubtedly reflects the stance of the club, who insist Suarez is not for sale and even if he was he is valued at much more.
Liverpool had already rejected the bid and plan talks with their star striker after Wednesday's friendly against Melbourne Victory, where he starts on the bench after completing just two training sessions having arrived in Australia on Sunday after an extended break following Confederations Cup action.
Reports have suggested a complicated contractual clause means the player can speak to any club offering more than £40m but the club insist that figure is not his buy-out price.
Press Association Sport understands when Arsenal made their first approach of £30m over a fortnight ago they were told by Liverpool that they would not entertain selling their prize asset even for £40m.
However, manager Brendan Rodgers and managing director Ian Ayre plan to hold talks with Suarez after the game as they attempt to persuade the 26 year old to stay despite his agent Pere Guardiola earlier this month indicating his client wants Uefa Champions League football.
Suarez will not be allowed to leave the club's tour early, with another friendly scheduled in Bangkok later this week before they return home.
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