The 25-year-old joined Swansea for £2.5m from Espanyol on an initial four-year deal in 2013.
The former Spain Under-21 international has made only two substitute appearances since Paul Clement was appointed head coach in January 2017.
Amat is the second Spaniard to return to his homeland this week following Borja Baston's loan move to Malaga.
Borja joined for a club record £15.5m from Atletico Madrid in August, 2016, but only scored once.
The 24-year-old almost joined Villarreal on loan in January, but the move fell through.