The Irons have managed just one win from 13 matches, although they are through to the quarter-finals of the Carling Cup where they host Manchester United.
Despite successive frustrating home draws against West Brom and Blackpool last week, Grant - who replaced Gianfranco Zola during the summer - feels his side are doing the right things to turn promising performances into maximum results ahead of their next game at Liverpool.
Co-owners David Sullivan and David Gold have both been supportive of the Israeli, whose reputation grew last season when he guided crisis club Portsmouth to the FA Cup final.
However, sections of the Upton Park faithful have made their feelings clear, as supporters again called for Grant's dismissal as boos greeted the final whistle following Saturday's stalemate.
Vice-chairman Brady, though, insists there is no chance of anyone wielding the axe too soon - maintaining blame cannot be laid solely at Grant's door, with the board having supported his summer spending plans.
Brady told BBC Sport: "We have not even had the discussion about whether Avram is going to be here or not be here - that is not for today, it is not for Christmas.
"It is unfair when you bring a manager in and things don't start exactly how you want - I think weak-minded people press the panic button
"We are not weak minded, but at the same time we did not spend the money we did in the summer just to survive
"We spent the money so we could feel comfortable, and we are not feeling very comfortable at the moment, but there are things we can do about that.
"It is not just all one person's fault - we can all do something, ourselves, the team and the manager.
"That is what we are focusing on, how we can work together to change the fortunes of the club."
The likes of Mexico winger Pablo Barrera, Denmark defender Lars Jacobsen and striker Frederic Piquionne, whom Grant worked with at Portsmouth, were all brought in during the summer transfer window.
Despite the club's massive debt issue, now around £85million, Brady maintains funds will again be available to strengthen Grant's options further during January.
"The transfer window helps to bring in new players which can change your fortunes," she said.
"We have been able to have a very good look to see where we think there are weaknesses in the team and that is our opportunity to go and put those weaknesses right, to bring in better players if we can attract them and find them because the January window is always the hardest to bring players in.
"We did quite a bit of business in the summer, but obviously not enough.
"We have to address that as a board and look very hard at where the issues are, put our hands in our pockets and bring in players."