The London club were flying high in the top four earlier in the season, but their campaign is in danger of finishing with a whimper as a result of their poor form which has left them without a victory since January.
Allardyce is out of contract at the end of the season and there is no sign that he will sign a new deal to remain at Upton Park.
Vice-chairman Brady has urged the experienced manager to mastermind a recovery ahead of Saturday's Premier League clash with struggling Sunderland.
She wrote in her column in The Sun: "Someone says in the press today that we flattered to deceive up to Christmas – but I 100 per cent disagree.
"The fact is that West Ham are underachieving and it is Sam Allardyce's task to identify the reason and rectify it. He would wish it no other way.
"I can almost hear Sam's post-match 'discussion' with the players [after defeat to Arsenal] from the boardroom. He wants and expects more."
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