Allardyce was appointed the Sunderland manager last week, and he is happy to be in charge of the Black Cats.
The Englishman had been without a managerial job since the end of the 2014-15 campaign when he left West Ham.
West Ham vice-chairman Brady has said that Allardyce did a good job for the London outfit, but she has warned that bad results could see the Sunderland fans turn against him.
“Sam Allardyce says he feels the love at Sunderland and I’m glad,” the 46-year-old wrote in her football diary in The Sun (Saturday, October 10 entry). “But he knows better than most that a couple of defeats can change that.
“He adds that wife Lynne used to get upset at her hubby’s social media messages from Hammers’ fans that were distinctly unsocial. I’ve received this garbage and sympathise.
"My opinion is Sam did a good job as our manager. I’m sure he’ll give Sunderland a boot up the backside. I hope the love lasts.”
Allardyce’s first game in charge of Sunderland will be against West Bromwich Albion away from home in the Premier League this afternoon.