The striker is currently with Uruguay on international duty at the Confederations Cup, but has made it clear during his time away from England that he would like to leave Anfield if the opportunity arose this summer.
Suarez said media intrusion was behind his attempts to bring an end to his Liverpool career, which began when he signed from Ajax in 2011.
Real Madrid are reportedly interested in landing Suarez this summer and the 26-year-old has admitted he would favour a move to Los Blancos.
Now Marca claim that Suarez has agreed personal terms with the Liverpool ace, who is expected to consider his options once Uruguay's participation in the Confederations Cup comes to an end, which could be as early as tomorrow.
Marca also claim that Suarez has offers on the table from other clubs, and that the Liverpool man will consider them if Madrid aren't forthcoming with a serious offer for his services.
The arrival of Carlo Ancelotti as Real Madrid's manager will go some way to clearing up the uncertainty regarding Suarez, as the Italian will likely firm up his transfer plans in the coming weeks.
Madrid would have to offer Liverpool around €47 million in order to activate the release clause in Suarez's contract.