The Zurich Premiership side need a stand-off after Italian star Ramiro Pez's move to Leicester and Hayes says he would jump at the chance to join them.
Hayes quit Warriors after failing to agree reduced terms for next season. He said: "I am in the process of applying for UK citizenship after living in Glasgow for more than four years and that would give me more flexibility.
"Rotherham are one of the clubs my agent has been in touch with and I would be thrilled to go and play in England.
"One of my other options is to head for Japan and, at this stage, I am keeping every avenue open.
"If my citizenship comes through quickly, it would obviously be a big selling point, along with the fact that I am not involved in World Cup matters and would be available throughout next season."