The Uruguayan is in talks with the Welsh club, according to reports.
The Uruguayan international has failed to make an impact at Southampton since joining the club in an eye-catching move back in 2012.
He was reported to have three main suitors for his signature, according to Tutto Mercato Web, Boro, Swansea and Bologna.
It was looking likely that Boro had won the race for the player, as reported by The Mirror last night.
Reports in Italy however suggest that Ramirez is in fact holding talks with Swansea City today - before deciding on his future.
The same report also claims he wanted a move to Bologna - but a deal is not financially viable at this moment.
He may have struggled in English football so far but Ramirez is clearly a player with bags of talent and hopefully he can find his feet wherever he ends up.