London Welsh are "seriously considering" leaving Oxford's Kassam Stadium, BBC Oxford reports.
Talks are understood to be under way between the club and "third parties" about staging matches in next season's Championship away from the city.
The club and stadium owners Firoka have so far refused to comment.
Welsh, who signed a three-year contract to play in Oxford in April 2014, were relegated from the Premiership last season after failing to win a game.
Supporters' club chairman Martin John believes any exit from the Kassam, which the club shares with League Two Oxford United, would be a result of dwindling attendances.
"I think everybody would have been a bit happier if crowds had been bigger last season," he said.
"The current stadium is a big one and we rattle around there really.
"We aren't getting enough people there and the costing of it must be quite immense."
London Welsh initially moved to the Kassam Stadium from Old Deer Park in Richmond in 2012 after winning promotion to the Premiership.
They continued to play there despite relegation back to the Championship in 2013.
Their on-field fortunes have continued to fluctuate since then and an exit from the Kassam Stadium has been discussed previously, as well as plans to redevelop Witney Town football club's former home as a 10,000 capacity ground.