The Hammers are believed to have seen their latest offer - said to be in excess of £30m - rejected, with Villa holding firm when it comes to their £40m valuation.
According to the London Evening Standard, West Ham are weighing up making another bid for the Colombia international, who has also been linked with Chelsea this summer.
Duran, 20, scored eight goals in 37 appearances for Villa during the 2023-24 campaign.
The forward arrived at Villa Park from Chicago Fire in 2023, and he has represented his current side on 49 occasions, scoring eight goals in the process.
Gil 'pushing' to complete Girona move
Meanwhile, according to transfer expert Fabrizio Romano, Tottenham Hotspur attacker Bryan Gil is pushing to complete a move to Girona this summer.
Gil, 23, made the move to Spurs from Sevilla in 2021, and he has represented the London club on 43 occasions.
The Spaniard has recently spent time on loan at both Valencia and Sevilla, and it is understood that he is pushing to secure a return to Spain's top flight his summer.
Romano claims that Gil 'has his heart set' on a move to Girona, who will play in next season's Champions League after finishing third in La Liga last term.
Personal terms have allegedly already been agreed, and the two clubs are now in negotiations over a fee, with the forward having another two years left to run on his contract at the Tottenham Hotspur Stadium.
Leeds 'handed Summerville boost'
Elsewhere, Leeds United are reportedly becoming increasingly confident of keeping hold of in-demand Crysencio Summerville during this summer's transfer window.
The 22-year-old was in impressive form for Leeds during the 2023-24 campaign, scoring 21 goals and registering 10 assists in 49 appearances in all competitions.
A number of clubs have been credited with an interest in the attacker, with West Ham and Fulham said two be among his admirers.
However, according to Football Insider, Leeds are sticking with their £35m valuation, and the club are yet to receive an offer near that price.
Liverpool and Chelsea are also said to be monitoring Summerville, who has managed 25 goals and 12 assists in 89 appearances for his current side in all competitions.