The veteran striker, 33, is now inside the final 18 months of his current deal and has been open about the prospect of Barca buying his replacement this summer.
However, Mundo Deportivo says Messi is insisting to Barca directors that Suarez be offered new terms. The Argentine now has a formal say on transfers courtesy of a clause in his contract and is demanding Suarez stay.
Just last week, Suarez declared: "My contract extension? I am very happy at the club. I always gave everything. Statistics and figures prove me right, I think I am up to it and when we have to talk, we should be okay."