The Uruguayan, whose Saints contract expires this summer, is on loan to Championship promotion-chasers Middlesbrough until the end of the campaign.
Ramirez moved to the Riverside in January, having played just 28 minutes for his parent club in this season's Premier League, and has scored three goals in his first eight matches for the Teesiders.
Earlier this month, the 25-year-old hinted at a permanent move to the North East, telling the Sun: "Why not stay? Football is about playing, which is why I moved here.
"I am enjoying this."
But with Boro's title challenge faltering amid uncertainty over Aitor Karanka's future as manager, a move to either Seville club is likely to appeal to Ramirez.
Southampton pulled off a coup by landing the former Bologna player upon their promotion to the Premier League, with ESPN reporting interest from both Liverpool and Tottenham Hotspur.
However, Ramirez has been restricted to just 55 appearances in his four years on the South Coast, and spent last season on loan to Hull City, which leaves an extension extremely unlikely.