Report: New Suarez bid rejected
The offer of £40million - plus a pound - was immediately refused by the Anfield club, it is understood.
Uruguay international Suarez is set to make his first appearance for the Reds this summer in the friendly against Melbourne Victory.
He jetted to Australia on Sunday after being allowed to take extended leave following his involvement with his country in the Confederations Cup.
While he was away the 26-year-old spoke about the attraction of Real Madrid, and his agent Pere Guardiola has told Liverpool the player wants to play in the Champions League but will not try to force through a move.
Suarez has been frequently linked with Arsenal, whose biggest spend on a player is the £15million they paid to Zenit St Petersburg for Andrey Arshavin in 2009.
By exceeding the £40million mark, Arsenal's offer means Liverpool have to inform Suarez of the offer but it does not trigger a buy-out clause.
Liverpool owner John Henry clearly wasn't impressed with the Gunners' bid, tweeting on Wednesday morning: "What do you think they're smoking over there at Emirates?"
The Reds had previously turned down a £30million bid from Arsenal, and manager Brendan Rodgers has said he "very much" expects the star man to be with the club come the start of the season.
Rodgers said on Monday: "There's been a lot of speculation over the course of the close season but the reality is that he's a player who is very much valued at Liverpool.
"And unless there's any sort of offer that comes in that's anywhere near his value, there's nothing to consider. We haven't had that."
Asked what that value was, Rodgers said: "I didn't say we would sell him. I said that every player has a value and a worth. Doesn't mean you've got to sell them."