The Uruguayan wants to leave the Riverside Stadium this summer after Boro were relegated to the Championship last season.
The 26-year-old was a target for Leicester City during the January transfer window, but he is now attracting significant interest from clubs in Spain.
Real Betis, in particular, are chasing Ramirez as they look to bolster their attacking options for the new season.
However the Spanish side have been dealt a major blow as, according to ABC de Sevilla, Ramirez has rejected their offer.
The playmaker has reportedly snubbed their approach as the move would force him to take a big pay cut from his current wage in England.