Hegyi has penned terms to 2027 and is also being sent on-loan to Motherwell for the new season.
Hegyi told the club's website: “I’m absolutely delighted to sign a new contract with West Ham. It’s been a long time since I arrived here as a kid, but London is like home for me now. I feel I’m progressing with every single season, so I’m really looking forward to seeing what the future holds.
“It’s great to know that the staff at West Ham believe in me. It’s a stable place for me, where I’m always happy, and I’m confident I can continue developing in a really positive way while I’m at the Club.
“The staff stayed in touch with me throughout my loans last season, and Xavi (Valero, first-team goalkeeper coach) was even coaching me through the phone. They really supported me, and I did my best to take on the advice they gave me.
“I think I took a big step forward during the loan moves, especially when I was in Holland, and as a young goalkeeper you just need to play games. Hopefully in the coming season I can continue on the same path.
“I’m really excited about the chance to join Motherwell on loan. It’s an opportunity to play in a big league, in some big stadiums, and I can’t wait to get on the pitch now and continue working hard.”