Wilshere, who only managed eight Premier League appearances in 2018/19 due to injury, feels a better start to the season can help West Ham challenge for the top six.
"That has to be our target, but the Premier League, you know what it is like, you have to fight every game, every week," the midfielder said after the Hammers' 1-0 defeat by Newcastle United in the Premier League Asia Trophy third-place playoff.
"Our problem last year was not against the big teams, it was against the other teams where we dropped points.
"We need to make sure we start right, last year we started badly, lost the first four games and then we are playing catch-up and it is difficult."
"I am not saying we are going to win the Premier League, but we want to push as high as we can.
"We beat Arsenal last year, we beat [Manchester] United, we beat Chelsea, so we know we can beat them.
"It's against the smaller teams where we need to focus more and pick up points."