But he said in a statement to Sky Sports News on Monday: "I do not wish to be considered for the position."
First team coach Kevin Keen and ex-Newcastle boss Chris Hughton are now thought to be the leading candidates.
Hammers co-owner David Sullivan has accepted responsibility for making a bad selection in choosing Grant as Gianfranco Zola's replacement one year ago and has vowed not to make the same mistake again.
told the Sun newspaper:
"Avram is a lovely person but the results sadly speak for themselves, that it was a bad selection by the board.
"We want a new manager that has a proven record of success.
"We will appoint a manager that will achieve our aims.
"I confidently predict that this time next year we'll be back in the Premier League."
However McClaren will not be the man to take on the task of securing an immediate return to the top flight.
The former England boss is without a club after being sacked by German club Wolfsburg earlier this season, following a successful spell in the Netherlands with FC Twente.
He admitted: "It is very flattering to be linked with a club of West Ham's size and historic stature, also one that has tremendous fans."
But as he distanced himself from the vacancy he added: "I wish everybody at West Ham all the very best for the future."
That appears to leave the Londoners, who BBC Sport understands want to make an appointment in the next two to three weeks, looking at other names on their wanted list.
It is thought that Keen, placed in temporary charge of the club, Hughton, Queens Park Rangers boss Neil Warnock, ex-Bolton, Newcastle and Blackburn manager Sam Allardyce and former Hammers defender Slaven Bilic are among the contenders.
Brighton & Hove Albion manager Gus Poyet has also been linked to the Hammers after leading the Seagulls to the League One title.
But he told BBC Sussex that he has not received an approach from the Premier League club: "It's something we're going to have all summer because of what I've done at Brighton. At the moment there is nothing at all.
"They're going to link me with plenty of jobs so I think we should stay calm. I'm concentrating on new players and cannot stop because of a rumour in the press.
"I'm not going to apply for any job. I'm in a great situation to stay at Brighton. If somebody wants me they need to approach Brighton."
Ex-West Ham left-back Julian Dicks, now in charge of non-League Grays Athletic, threw his hat into the ring - although he conceded it would be a big step up.
NEXT WEST HAM BOSS ODDS
Source: William Hill on 16 May
"Of course it is but that's not saying I can't do the job," he told BBC Radio 5 live. "I'd love to have the chance to do it."
While Sullivan spoke of the need to appoint a manager with a proven track record, former West Ham striker Tony Cottee told BBC Radio 5 live that he is concerned the club could struggle to pay the salary that a top-quality manager would command.
"I can't see the current board offering big wages to any manager," he said.
"My top choice for the job is [former Aston Villa manager] Martin O'Neill but I don't know whether he would want to manage in the Championship and he would want big money but he is the standout candidate.
"The owners have a decision to make but they will have to pay money to get a good manager to get the club back in the Premier League."
For Keen, it is a third spell as caretaker manager since the departure of Alan Pardew in 2006 and he will pick the team for West Ham's final game of the season at home to Sunderland on Sunday.
However, the Hammers will have only pride to play for following the 3-2 defeat at the DW Stadium at the weekend.
Gold stated: "This is undoubtedly the worst moment in all the years I've been supporting West Ham.
"I honestly believed with the players we brought in in January and the imminent return of [Thomas] Hitzlsperger, we had done enough to pull clear of danger.
"I know that Scotty Parker shared that belief with me and we both had confidence that we would retain our Premiership status."
Gold told Sky Sports News that the club would "start a whole rebuilding programme" and added: "We have to prepare a squad of players that is going to get us back into the Premier League at the first time of asking.
PHIL McNULTY BLOG
“Grant's sacking puts him in the dock for a season of managerial folly at Upton Park but he will not stand alone when the inquest gets under way”
"West Ham is a great club with a great tradition. We're entitled to grieve but sooner or later we have to stop feeling sorry for ourselves and start believing and rather than talking about relegation we have to talk about promotion."
Former Chelsea and Portsmouth boss Grant took over from Gianfranco Zola in June 2010 on a four-year contract.
But after beginning the season with four straight defeats, the east Londoners were always in a relegation battle, despite the efforts of Parker, who was named
Football Writers' Player of the Year
Needing a win at Wigan on Sunday to stand any chance of staying up, Grant's side went 2-0 ahead before collapsing in the second half and eventually losing to a Charles N'Zogbia winner four minutes into stoppage time.
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