The Times claims that if Glasgow do have to go outside the city, the new stadium at Falkirk is the most favoured option.
Gordon McKie, the SRU chief executive, has revealed that five grounds are being examined to see if they could play host to the Warriors next season, but added that there are problems with all of them.
He refused to identify which grounds have reached the shortlist, but it is understood that they are the rugby club grounds at Anniesland, Burnbrae and Hughenden, the football ground at Firhill and the community stadium at Falkirk.
Forthbank and Broadwood stadiums have been mentioned in the past but they do not feature in the union's present planning.
The difficulty for McKie is that none of the rugby grounds have good enough facilities and going to any of them would involve extra costs for the cash-strapped union.
To become a season-long home, Burnbrae would require temporary floodlights and seating, as would Anniesland, to satisfy expected crowds of 3,000 to 4,000 and safety regulations, while Hughenden needs significant improvement work in its changing areas
Firhill is the best option inside the city in terms of facilities but there were problems when the pitch was cut up by rugby players the night before Partick Thistle, the owners, played there, and it was never popular with the Glasgow fans.
Falkirk's modern pitch is designed to take the pounding of a rugby game and still be in perfect condition for a football match — but it is 20 miles by road from the city's north-east boundary, and many Glasgow fans would be likely to be deterred by the journey.
"Obviously, we are a rugby business and it would be nice to play in a rugby stadium," said McKie in today's Scotsman, "but there are few that fulfil the criteria we want.
"Falkirk is being evaluated as it has a good pitch and floodlights. We've used it for Under-21 games and it is accessible. Clearly, that move (so far from Glasgow) would be emotive and our preference would be to stay in Glasgow. Sadly, not many locations in Glasgow can tick many boxes in the criteria we have to host professional rugby."
In the long term, he gave warning that the future of both Glasgow and the Borders is still being assessed on a season-by-season basis. While the strong view within the union is that there should be three professional teams, there are no guarantees about where they should be based.
"There is no five-year plan for either of them," McKie said in The Times. "We reassess where we are at the end of every season. We have shown in the sale of Edinburgh what can be achieved and ideally that will be the model for the other clubs as well, but things appear to have gone a bit quiet on that front."