Grant favourite for Upton Park job
The east London club, who finished one place above the Premier League relegation zone, have been without a manager since Italian Gianfranco Zola was sacked on May 11.
NEWS: Grant departs Portsmouth
"West Ham United can confirm the club is making good progress in the search for a new manager and the appointment will be made by early June," the Upton Park club said in a statement on their website on Friday.
"The club is in the middle of a thorough selection process which could continue over the next 12 days before being finalised."
Former Chelsea boss Grant, 55, is expected to get the job after resigning as manager of relegated Portsmouth on Thursday.
The Israeli defied the odds in guiding the cash-strapped south coast club to this season's FA Cup final where they were beaten 1-0 by Premier League champions Chelsea.