The Hammers finished in a disappointing 11th place last season, but have brought in the likes of Hart, Pablo Zabaleta, Javier Hernandez and Marko Arnautovic this summer in a bid to improve that showing.
West Ham will begin the new campaign with a tricky trip to face Manchester United at Old Trafford, but Hart is looking forward to getting started having spent the previous season away from the Premier League with Torino.
"Our intention is to be inside the top eight in the Premier League and even higher than that. The chairmen have backed the team with the money that they've spent and we have to aim high," he told Sky Sports News.
"Realistically, winning the league is going to be extremely tough, but we're going to aim for the highest we can do and compete at every level. We've got an interesting start to the season, away at Manchester United, and that will be a good test for us.
"I can only speak for myself, I have still got a lot left in me. I love football and I just want to play. I am so grateful to be given the opportunity to try and play at West Ham. I am going to train hard every day and push myself. I have obviously got all those experiences, I've got the titles and medals [at City], but I want more and if we can do that here, I am going to give it all that I've got."
West Ham will face Werder Bremen, Altona and Manchester City in the remainder of their pre-season preparations.