Liverpool city council leader Mike Storey has admitted it is unlikely there will be a Merseyside groundshare.
Liverpool have consistently ruled out sharing with Everton
Everton and Liverpool are set to make a final decision this month.
Liverpool have planning permission for a new 60,000-seater stadium, while Everton are meeting council leaders to discuss improving Goodison Park.
Storey told BBC Radio Five Live: "I guess that sadly a groundshare won't happen and in years to come we will regret it."
Everton chairman Bill Kenwright has warned he would resign rather than be tenants of Liverpool's new stadium.
And Storey said: "Everton are right in saying it would not work if they were tenants. It would work if both clubs had a stake."
Officials from Everton and Liverpool met Sports Minister Richard Caborn, who has asked them to consider sharing.
But Liverpool have already insiststed they are pressing on with their new stadium in Stanley Park, despite costs rising from £80m to £115m.
Everton previously had plans to build their own new ground at King's Dock but that was shelved because of the cost.
Liverpool manager Rafael Benitez has insisted that a groundshare would be a bad idea.
"Talk of two teams sharing a stadium always causes problems and in my view it would not work," the Spaniard told his club's website.
"It would not be good for the pitch because obviously there would be double the amount of games played on it and it would not be good for the supporters of Liverpool or Everton.
"True, Inter and AC share a stadium in Milan but look at their pitch. It is not good. Also, in terms of atmosphere it is better that a team plays in its own stadium.
"In five years time we hope to be playing good football in our own new stadium."