During the summer, Swansea decided to pay a fee of £11m in order to sign Mesa from Las Palmas, but the move is yet to work out for the 28-year-old.
Mesa has made 16 appearances in all competitions but his last outing in the Premier League came in Carlos Carvalhal's first game in charge on December 30.
According to AS, Sevilla are keen on handing Mesa an escape route during the second half of the campaign, providing that Swansea agree to pay a portion of his wages.
New manager Vincenzo Montella is assessing his options in the transfer market, and it appears that he was impressed with Mesa's previous spell in Spain.
During the last two campaigns, Mesa made 69 appearances in La Liga for Las Palmas.