The Ecuador international joined the Liberty Stadium outfit in 2014 and has played 72 times for the Swans since signing from Mexican outfit Monarcas Morelia.
However, he has become surplus to requirements under Paul Clement, making just 14 appearances during an injury-hit 2016-17 campaign, and has yet to feature in a matchday squad this season.
Montero's most recent Swans appearance came in the 1-1 draw at Manchester United in April, when he arrived off the bench in the second half only to be forced off with injury seven minutes later.
Swansea confirmed the move on their official website, stating that Montero's "fitness troubles have been a frustration for the Swans, because Montero is capable of being a significant attacking threat when he is firing on all cylinders".
The transfer window in Spain does not close until midnight on September 1.