The Hammers have been linked with a host of top strikers this summer and are believed to currently be focusing their attentions on AC Milan's Carlos Bacca.
With the big move to the 60,000 seater Olympic Stadium on the horizon in the coming month speculation has been rife about new additions to Slaven Bilic's squad.
The East Londoners missed out on Marseille's Michy Batshuayi after he secured a move to London rivals Chelsea while there were also rumours about interest in Bournemouth's Callum Wilson, Juventus and Italy forward Simone Zaza and Liverpool's Christian Benteke to name but a few.
Lacazette has a fearsome scoring record
Alex kid was West Ham's main target
Bilic's most concrete interest until now, though, appears to have been in France forward Lacazette with their club record bid turned own last month, as reported by Sky Sports.
Given the Lyon star's age - he is just 25 - potential and goal ratio of just under one in every two matches, many Hammers fans have been hoping the club can push through the marquee striker signing their owners David Sullivan and David Gold have been craving.
Hammers kiss their hopes goodbye?
'We are very happy'
"It seems Alex wants to spend this season with Lyon in the new stadium," Aulas told L'Equipe.
"He wants to know an adventure in the Champions League could be much better than last season.
"I think he will stay with us and we are very happy."
Lyon's new home
Swanky new ground not as big as Hammers' Stratford home
Lyon's new 59,186-seater Parc Olympique Lyonnais, nicknamed the Grand Stade and the Stade des Lumières, is not quite as big as West Ham's new Olympic Stadium however which is 60,000 seater and will increase to 66,000 from the 2017/18 season.
It replaced the club's previous stadium, Stade de Gerland, in January and next season will be the Ligue 1 side's first full campaign in the swanky new ground.
Arsenal manager Arsene Wenger
Wenger circling too?
Unsurprisingly given Arsene Wenger remains the manager at the Emirates, Arsenal have also been credited with an interest in Lacazette.
So if he was to decide on a move to the Premier League, Bilic would face stiff competition from North London for his signature.
Wenger's somewhat uncertain long-term future could give the Hammers the edge, though, should the situation arise.