Los Che's major financial concerns have led them to admit that they may be forced to part with their most valuable assets in a bid to ease their troubles.
Spain international Villa signed a six-year contract with Valencia in August which ties him to the club until 2014.
But the Primera Liga outfit have now held initial discussions with City over a big-money move for the 27-year-old, although the Eastlands outfit baulked at the initial valuation.
Despite the financial backing of the Abu Dhabi United Group and their failed £100million bid for Kaka still fresh in the memory, the Premier League club apparently do not want to pay over the odds.
Significant amountWhen asked if he could confirm that Valencia chiefs had met with two representatives from Manchester City regarding the sale of Villa, Gomez said: "Yes, it's true but they didn't make us any offer.
"The president (Vicente Soriano) asked for a significant amount and they didn't agree."
Gomez did not reveal in the interview on Spanish radio what price tag they had put on Villa and his fellow Spain star David Silva, but insisted they would not be short-changed for either player.
"I think they are two very important players, with a lot of quality and, if we sell them, we are not going to sell them on the cheap," said Gomez.
He added: "I would like them both to stay. But we cannot discard anything and any of them (the players) could leave at the end of the season."