The Colombian international is understood to be interested in returning to European football, although Vasco are determined not to make his departure straightforward.
Bournemouth have shown the strongest interest so far and have maintained contact with Vasco over a possible deal. The English club, who signed Rayan from the Rio de Janeiro side last year, have been discussing Gómez for some time, with negotiations expected to gather momentum around the turn of the year.
Everton and Nottingham Forest have also made enquiries about the 24-year-old, although neither has progressed to more detailed discussions. Their interest gives Vasco several potential destinations to consider, but Bournemouth currently appear to be the most advanced of the three clubs.
Forest have an established relationship with Vasco through previous transfer dealings, including the loan of left-back Cuiabano. However, there has been no indication that the defender could be included in a possible transaction for Gómez.
Vasco are aware both of the growing interest from England and of the player's ambition to return to Europe. Nevertheless, the club believe they are in a strong negotiating position and are targeting a fee of approximately €20 million (£17.4m).
That figure would represent the second-largest sale in the history of the Brazilian club. It would also be a substantial return on their investment after Vasco acquired 60 per cent of Gómez's economic rights for €4.5 million (£3.9m), equivalent to approximately R$27.78 million at the current exchange rate quoted in Brazil.
Gómez has made 72 appearances for Vasco, scoring nine goals and providing 14 assists. His performances have established him as one of the club's valuable assets, while his international status with Colombia has further increased his profile.
Vasco's valuation is also influenced by the circumstances of his arrival. The club invested in a significant share of his rights and will want to secure a fee that reflects his contribution and potential resale value.
Bournemouth's previous business with Vasco could help the clubs maintain a constructive relationship, but the Brazilian side have made clear that they will not simply allow one of their important attacking players to leave at a reduced price.
The interest is expected to intensify as the next transfer window approaches. For now, however, enquiries from Everton and Forest remain at an early stage, while Bournemouth are the club most closely associated with a possible move.
Vasco's expectation is that any agreement will need to come close to their €20 million target, giving them the opportunity to turn Gómez's departure into one of the biggest sales in their history.