The Hammers have already added Mauro Zarate, Cheikhou Kouyate, Aaron Cresswell and Diego Poyet to their squad in the off-season and joint-chairman David Sullivan told supporters to expect a fifth arrival in the near future.
Valencia, who hit 18 goals in 23 appearances for Mexican side Pachuca last season, impressed at the World Cup – scoring three times in as many games - and the 25-year-old is nearing a £12million switch to Upton Park, according to Sky sources.
Allardyce is looking to improve on last season’s 13th-place finish in the Premier League and he confirmed that another signing looks to be on the horizon.
He told Sky Sports News: "We're very close, but nothing is ever certain in this game with new signings until they have actually signed on the dotted line.
"We are very hopeful, there has been a lot of negotiation done by the chairman and hopefully that will pay off in the next 72 hours."
West Ham got their pre-season fixtures underway with a 2-2 draw at Stevenage, but they were without record signing Mauro Zarate and fellow striker Andy Carroll as late goals from Matt Jarvis and Stewart Downing saw them recover a two-goal deficit at Broadhall Way.
The club played down the absence of Zarate and Carroll and say the pair will be fit to fly to New Zealand next week as preparations for the new campaign are upped, but Allardyce will also be hopeful of having Valencia on board.