The pacey winger attracted attention from Cardiff over January, but the Bluebirds were unable to tie up a deal despite strong interest.
Since then the Swans have been taking notice of the Monarcas Morelia ace, who has been earning rave reviews for his displays in the Mexican game.
Montero value their man at around £4m, and Swansea are said to be keen to pay such a fee before the World Cup starts – where they fear his stock may rise id he performs well.
Swansea attempted to land Tom Ince in January, and are still open to idea of bolstering their wide options.
It is understood that the South Wales outfit are ready to open talks with Morelia, and are hopeful that a deal can be agreed swiftly.
Despite being just 24-years-of-age, Monetro has already enjoyed one spell in European football, playing for Villarreal, Real Betis and Levante – the latter two were loan deals.
After failing to establish himself in Spain, he moved to Mexico in 2012, where he has thrived.