Grant's side are bottom of the Premier League on 20 points
West Ham are "committed to retaining Avram Grant as manager" after reports had linked Martin O'Neill with his job.
BBC Sport understands that O'Neill, who left Aston Villa in August, decided over the weekend against taking up any offer to succeed the 55-year-old.
West Ham are bottom of the Premier League but have won three of their last six games under Grant.
The club said in a statement that "potential transfer targets" had been identified in a bid to beat the drop.
Allegations in Tuesday's Daily Mirror newspaper that vice-chairman Karren Brady had asked some of the players to "rise up against Avram Grant" were also categorically denied.
The Hammers said they intended to find the source of what they called "unhelpful and untrue" comments and would remove whoever was responsible from the club. They added that their lawyers were now dealing with the matter.
During his eight-month spell in charge at Chelsea between September 2007 and May 2008, Grant came within a penalty shoot-out of delivering the Blues' first Champions League trophy and his West Ham side have also excelled in knockout football.
They will take a 2-1 advantage to St Andrew's for the second leg of their Carling Cup semi-final against Birmingham and made the fourth round of the FA Cup with a 2-0 win over Championship side Barnsley.
Although money may be made available to Grant, his purchases while at Upton Park have only been of limited success.
Striker Frederic Piquionne has scored seven goals since his arrival from Lyon, but Mexico winger Pablo Barerra and New Zealand defender Winston Reid have struggled to make an impact.
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