Villa, Fernando Torres' partner in attack for the Spanish national team that became European Champions last summer, was manager Mark Hughes' main target in the January transfer window ahead of the failed attempt to sign AC Milan's Kaka for a world record fee.
However, with clubs all too aware of City's vast wealth since they were taken over by the Abu Dhabi United Group, the fee demanded for the player led to negotiations coming to nought.
According to a report in the Guardian newspaper, City opened talks with Valencia over a combined £100m bid for the two players, only for the Spanish club to request a minimum of £135m.
Valencia are straining under the weight of enormous debts, however, and Gomez has given the clearest indication yet that the club may be forced to cash in on their most saleable assets.
Villa is believed to favour a move within Spain, or remaining at Valencia, ahead of a transfer to the Premier League, despite interest in the past from Chelsea and Liverpool.
Should City table an acceptable offer, however, Valencia may feel pressure from his current employers to consider such a move.
'The club is in a very delicate situation,' Gomez said. 'It has to control spending, grow income and sell assets.'
'Obviously, we will consider [player sales]. We have to control costs and the biggest cost in a football club is maintaining a team.'
Gomez was appointed this week and his taking of office in itself
increases the possibility of players leaving.
He was backed by Bancaja, the Spanish bank who are the club's major creditor, being owed €240m of Valencia's reported debts of €450m.
The financial difficulties of the club has led to players going unpaid and work on a new stadium, to be funded by the selling of land around their current home, which was scheduled for completion in May, has come to a standstill.
'Before, we had a plan that was based purely on selling the land – now we need to seek alternatives,' Gomez said. 'We need to win back credibility with the financial institutions.'