Maynard has been strongly linked with a switch to Elland Road after failing to secure a regular first-team place following his move from Bristol City back in January.
The summer arrival of Modibo Maiga appeared to push him even further down the pecking order.
But he was handed a rare starting place against Crewe in the Capital One Cup on Tuesday night and impressed Sam Allardyce by getting on the score sheet during the 2-0 victory.
Maynard is now hoping to keep his place for the London derby against Fulham on Saturday and says he wants to remain at Upton Park.
Maynard said: "My friends have made me aware of the speculation but I am happy to stay at West Ham and it is all about competition for places.
"I think the gaffer likes to have a lot of options and for all the strikers that come in it will make us raise our game.
"I would have liked to have got a few more starts and a few more goals, but I feel I contributed and just hope I can get another chance to prove my value to the team. Any player wants to compete in the Premier League and I am no different."