Mesa joined Swansea from Las Palmas for a fee of €12.5million in July after two excellent seasons in La Liga.
The 28-year-old has started just nine of rock-bottom Swansea's 24 Premier League games this season, however, with the last coming on December 30 against Watford.
Mesa had been linked with a return to Las Palmas but Sevilla, reportedly keen on signing a back-up to Ever Banega, have won the race and will be able to offer him a first taste of Champions League football.
Vincenzo Montella's side face Manchester United in the last 16, with the first leg at the Ramon Sanchez Pizjuan taking place on February 21.
Sevilla have been long-term admirers of Mesa and had been particularly keen on him when Banega left for Inter in 2016, though they felt they were priced out of a move.