The 24-year-old was impressive for his nation during their three group games in Brazil, and has been on the radars of Premier League clubs for some time.
Newcastle and West Ham are understood to be tracking the pacey wide-man, but Swansea are confident of winning the race.
Montero currently plies his trade in Mexico with Morelia, but has featured In European football earlier in his fledgling career.
He failed to make a major impact as a youngster with Villarreal, Real Betis or Levante, but has thrived since heading to Central America.
The winger’s employers are understood to be ready to consider serious offers for Montero, who himself admitted recently that he is hoping to move on:
“The club are handling it. Let’s see where I’ll go but for now I have some days off,
“It’s certain that I’ll leave the (Mexican) league, but let’s see where I’ll go. We expect to sort it out within the next few days. It will be Europe and thank God we are working on it, I hope it works out.”