Sullivan and Gold were approached in February by a consortium which included former QPR chief executive Philip Beard.
But West Ham turned down the approach because Sullivan and Gold felt it was not in the best interests of the club or the supporters, and the pair insist they would only consider a sale were any prospective buyers able to fulfil that criteria.
West Ham also say the consortium that approached them did not provide proof of funds.
While there has been no formal offer to buy West Ham, the Daily Mail reported on Wednesday that the proposed approach totalled to £400million.
Premier League clubs, including West Ham, are often the subject of informal approaches by interested parties.
Sullivan is open to external investment in the club but does not want to lose a controlling stake.
In 2016, his son Jack claimed a bid of £650m for West Ham from Red Bull had been rejected.
Interest from an American consortium was also rebuffed last year.