The Israeli rejoined the Hammers on loan from Chelsea in the summer but is yet to start a league game for Sam Allardyce's side.
Ricardo Vaz Te's shoulder injury could allow Benayoun the chance to secure a first-team spot, and the playmaker admits he is keen to establish himself in the West Ham side.
Benayoun's current loan runs until January but he hopes he can persuade Allardyce to take up an option on his deal and secure a stay until at least the end of the current campaign.
He told talkSPORT: "West Ham is one of my favourite clubs and it's great to be here. Although it is only a loan, there is an option in January to extend it to the end of the season.
"My ambition is to be at the club for a full season and to make it a good one.
"There is strength in depth here. When I was here the first time we had a few talented players but this time we have a lot of quality - Mark Noble, Matt Jarvis, Kevin Nolan and Ricardo Vaz Te.
"I think we have a nice balance between attackers and defenders, which helped us to start with three clean sheets in four league games.
"If you can do that in the Premier League, it shows the strength of the team defensively and we have the talent offensively to score goals."