The English League One club are looking to replace Luke Williams who left the Wiltshire outfit last week after they were relegated.
Archibald has impressed The Robins owner/chairman Lee Power in guiding Thistle to a top six finish in the Premiership this season.
Partick Thistle say they have yet to receive an approach.
Archibald was approached by Shrewsbury in October but decided he had "unfinished business" with Thistle.
The 39-year-old is the longest-serving manager in Scotland's top flight and was in the four-man shortlist for the PFA Scotland manager of the year, won by Celtic's Brendan Rodgers.
Other names, who have been linked with the Swindon vacancy but appear to be out of the running, are the former Queens Park Rangers boss Jimmy Floyd Hasselabaink and Mansfield Town manager Stevie Evans.
Partick Thistle manager Alan Archibald is on a shortlist of candidates for the Swindon Town vacancy.