Nunez is highly rated in Portugal due to his exploits for Benfica and the Lisbon side have seen clubs show interest in snaring him away this month.
It was claimed earlier this month that Newcastle, who have been looking for a striker, were prepared to bid for Nunez, but the trail has gone quiet in recent weeks.
And it is West Ham who are now leading the charge to take the forward to the Premier League before the transfer window shuts.
It is suggested they are pushing for a deal, with one source claiming that the Hammers have gone as far as agreeing a fee with Benfica.
However, the swoop being left until late in the transfer window means doing the deal could be tough as Nunez is in South America for Uruguay’s World Cup qualifiers.
Newcastle have just had a team in South America to push the capture of Bruno Guimaraes over the line.
Whether West Ham can find a way to put Nunez through his medical paces before the transfer window shuts remains to be seen, as does whether further sides will enter the race.
