After the final day of the season, Neves suggested that he was open to taking on a new challenge away from Molineux after five years in the West Midlands.
The Portugal international has been consistently linked with the likes of Barcelona and Manchester United, the former said to be the 25-year-old's preferred destination.
However, the Catalan giants are currently not in a position where they can fund a move Neves, whereas United only see the player as one of several different midfield targets.
According to Jornal di Noticias, a scenario has materialised where Neves could opt to pen fresh terms rather than push for a transfer elsewhere.
The report claims Wolves could hand their most prized asset a contract worth €4m (£3.43m) per year after tax, an increase on his current earnings at the Premier League side.
With Neves's contract having just two years left to run, Wolves find themselves in a position where their key player's valuation would plummet should they allow him to enter the final 12 months of his deal.
Wolves allegedly have no intention of accepting any offers below €50m (£42.93m), a stance which does not suit Barcelona and may result in United looking elsewhere.
Bruno Lage will want a resolution either way as soon as possible, the Portuguese head coach currently waiting to see whether the club can deliver at least one of two midfield targets.
Earlier this week, it was suggested that Wolves are prepared to break their transfer record in order to win the race to sign Sporting Lisbon midfielder Matheus Nunes.
However, the assumption has been that the 23-year-old would arrive at Molineux as a direct replacement for Neves, rather than the pair joining forces in the centre of the pitch.
Wolves are also long-term admirers of Joao Palhinha and it appeared at one stage that the 26-year-old moving to the English side was a foregone conclusion.
Lage's men have seemingly since dropped behind Fulham, the newly-promoted side already said to have agreed a fee with Sporting and personal terms with the player.
Although Wolves are yet to make a first-team addition this summer, the likes of John Ruddy, Marcal and Romain Saiss have already left on free transfers.