The 24-year-old, who represented Ecuador at this summer's World Cup in Brazil, has arrived from Mexican club Morelia and becomes the Welsh side's fifth signing of the summer.
The 24-year-old joins the Premier League from Mexican side Morelia and has signed a four-year contract with Garry Monk's side.
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Montero trained with his new teammates on Thursday and becomes Swansea's fifth signing of the summer after the arrivals of Bafetimbi Gomis, Stephen Kingsley, Marvin Emnes and Gylfi Sigurdsson.
The former Villarreal winger has been capped 43 times by Ecuador and started all three of group matches at this summer's World Cup in Brazil, but could not help the team reach the knockout stages.