They could face competition from Everton, who are also reportedly interested in the Ecuadorian.
The Hornets appear to be keen on adding some pace to their wings in the January transfer window, having already been linked to several wingers.
One of those is Tottenham man Andros Townsend, with the Guardian reporting they have already had a £10million bid turned down for his services.
They are unlikely to offer more than that, and instead have now turned their attention elsewhere with them allegedly keen on Swansea man Montero, according to a report from the Mirror.
The Ecuadorian started the season in fine form for the Swans, but he has not been a guaranteed starter in recent months and may fancy the move.
The Welsh side are set for a relegation battle this year, whereas Watford have impressed and are currently 9th in the table, just three points outside the top six.
Whether Swansea are willing to sell is as yet unknown, although Watford could face competition for his services regardless, with Everton also reportedly interested in adding the winger to their squad in January, according to the Sun.