The 33-year-old Wilson has had 111 direct goal involvements in 239 Premier League appearances, including 49 in 130 games for Newcastle, where he failed to agree a new contract.
He has played nine times for England, scoring twice.
"I'm excited to be joining a massive football club like West Ham. A club that shares a similar passion to mine, hard work, dedication and also I strive for success so I'm glad to be a part of it," Wilson said.
Wilson moved to Newcastle in September 2022 and had his best Premier League tally of 18 goals in 2022-23, a season that also saw him feature for England at the World Cup in Qatar.
He played in just 18 league games last term due to injuries.
"He is a proven, experienced Premier League striker, with a great work ethic and a fantastic goalscoring record," West Ham coach Potter said.
"He also has excellent character and personality, which is such an important factor, and something that we place a big emphasis on with our player recruitment."