As it stands, the Irons sit fifth in the Premier League table, one point behind fourth-placed Manchester City with eight games to go as the dream of Champions League football looks realistic.
Finishing fifth would guarantee them a spot in the Europa League, and even that would represent a remarkable first season in charge for Slaven Bilic, who replaced Sam Allardyce last summer.
Their top-four bid took a slight knock in Saturday's 2-2 draw at Chelsea, in which the hosts were awarded a late penalty for a foul that appeared to come outside the box, as the Blues scored and denied West Ham all three points.
“We are there waiting to pounce should the opportunity present itself and if we finish in the top five, then that will be a great achievement for us," Payet told the club's official website.
“At Chelsea, we were a couple of minutes or a few metres from the victory. For their first, we felt that the wall was too far back and for their second, it seems that the foul happened outside of the box.
"This is why we are disappointed with the result."
With West Ham set to leave Upton Park after a 112-year stay this summer, European football of any kind would be a fantastic way of celebrating their move to the Olympic Stadium.
They were involved in Europa League action this season, but Bilic's side failed to reach the group stages of the competition, suffering elimination in the preliminary stages.