West Ham have now confirmed the deal for the 30-year-old for an undisclosed fee after he spent 18 months with Steve Bruce's side.
Jelavic told the club's official website: "I am very excited to be signing for West Ham United, especially as it is a big club and I didn't know until a couple of days ago that I would have the opportunity to come here.
"Of course it is always a target to play in the Premier League, so when I found out that West Ham wanted me there was no doubt that I wanted to come here.
"I've played for Slaven before for the Croatia national team and he is a great manager and a great man. I am really happy and delighted to be working with him and all his staff again. I'll be joining a team that's on a high after the result on Saturday."