Press Association Sport reports that the Gunners made an offer in the region of £30 million for the Uruguay international last week which was rejected out of hand by the Reds, while the BBC has made an identical claim.
Despite the 26-year-old this summer voicing his dislike of his treatment by the English media and suggesting a move to Real Madrid would appeal, the club have no intention of selling, a stance which was not even tested by an offer well below what is considered Suarez's true value.
Suarez, who has three years remaining on an improved contract he signed last summer, is currently on extended leave after his involvement in the Confederations Cup.
Neither he nor his representatives have spoken to Liverpool about any intention to ask for a move but his comments while on international duty in South America have left the club in a difficult position.
It was expected any approach from Madrid would come after the end of the Confederations Cup but so far has nothing has materialised and the fear is the Spanish giants will leave it late in the window in order to drive the price down.
Arsenal's surprise bid may have been more strategic than realistic as they are currently trying to push through a deal to sign Real striker Gonzalo Higuain.
Making an offer for Suarez may have just been a ploy intended to focus the minds of Bernabeu officials on signing off the Higuain move.
Suarez is not due back for pre-season until July 22 and he will then be expected to join the club's tour matches in Australia and Thailand.