The Israeli tactician will sign a four-year deal to become Gianfranco Zola's successor and should be unveiled this week.
In what has been one of the worst-kept secrets in recent times, the Hammers will get their man who took Pompey to the FA Cup final despite the fact the club was falling down around him and this appealed to West Ham's co-owners David Gold and David Sullivan.
Gold said yesterday: "After an hour (of meeting Grant), we knew we could work with this manager. We are very optimistic.
"It's time for the football club to get some stability with people that have been there, done that and got the T-shirt. The relationship between the manager and the chairman is critical.
"In our case it's two chairmen so it's even more difficult.
"What we need is a manager who has achieved, has experience. We are not in a position where we can take on young, inexperienced managers."
A statement on the club's official website said: "West Ham will make a formal announcement of the club's new manager within 24 hours.
"The club have been conducting a thorough recruitment programme and made excellent progress.
"The squad are set to report back for pre-season training on July 5, meaning the new manager will have plenty of time to put together his squad."