The Reds striker was a target for Los Blancos last summer and their interest has only intensified after the Uruguayan man hit 31 goals last term en route to becoming the double Player of the Year.
The Champions League winners were reportedly preparing a world-record bid for Suarez, but the former Ajax star has told his teammates that he has not intention of leaving Anfield.
According to the Daily Mail Suarez is staying to help the Reds challenge for the Premier League title next term after falling short to eventual winners Manchester City last time out.
Suarez's tears at the end of their 3-3 draw to Crystal Palace earlier this month was seen as a sign by some that the forward had seen his chance of title win in England gone.
However, the report claims that sources inside the club have been assured Suarez is committed to staying at Anfield and has begun plans on enrolling his daughter Delfina in a school in the Merseyside area.
Suarez became the highest-paid player in Liverpool's history when he signed a new contract in December last year to complete a remarkable turnaround after looking odds-on to leave the club last summer.
Following his 10-game suspension for biting Branislav Ivanovic, Suarez gave a number of interviews to a national newspaper in an attempt to engineer a move away.
He was banished from the first team by manager Brendan Rodgers before eventually coming back into the fold to equal a Premier League record held by Alan Shearer and Cristiano Ronaldo for number of goals in a 38-game season.