Fallon is out of contract at Swindon in the summer, but has been offered a new deal to stay at the County Ground, although he is yet to sign.
It's thought that Swansea have up to £400,000 to get a new striker, but is not known how much the bid Swindon rejected was for.
Mark Devlin told the Swindon Adver: "All I can say is that we have received an offer and it's been rejected. We are hoping Rory will sign a new deal with us."