The footballing authorities responded to Brady's statement with surprise, but may be powerless to prevent such a change.
Birmingham are desperate to return to football's top division which they left - when it was still called the First Division - in 1986. To do so Brady is apparently ready to explore every avenue, including dropping a name which has served them for nearly a century.
Speaking on BBC2's Working Lunch, she said: "I would be prepared to change the name of the club and I think that is the way most clubs are going.
"You cannot do everything on your own. We do need a partnership with someone whose ambitions match ours. We took 48,000 people to Wembley recently, and it could have been more, which shows our potential to a big company.
"We would change the name, either of St Andrews or Birmingham City. That would be something which is going to happen with such partnerships in future - and I hope it happens."
Birmingham were founded as Small Heath Alliance in 1875. In 1888 the "Alliance" was dropped, and in 1905 they became Birmingham. Forty years later the "City" was added.
Both the Premier and Nationwide League said yesterday that there was nothing in their regulations to prevent a club changing its name to include a sponsor. However, the Football Association stressed that any proposal would need its approval.
A Football League spokeswoman admitted the idea was "untested water" but remained open-minded. She said: "We have had name changes in the past, such as Orient becoming Leyton Orient again, but adding a sponsor has no precedent.
"There is nothing in the regulations which suggests we would automatically disapprove of such a change. One concern is it might get confusing were clubs to change their name every time they changed their sponsor which at the moment is quite a frequent occurrence."
Mike Lee, a Premier League spokesman, said: "It is an interesting question, but none of our clubs are proposing such a switch so it not an issue. As it stands however there is nothing in our rule book to prevent a name change, but if a club were planning to do so formally we would expect to be consulted as would the FA."
Steve Double, the FA spokesman, said: "At county level incorporating a sponsor's name needs FA approval, and that would apply to any league club too. There are a series of rules which govern sponsorship so such a major step would need to be sanctioned - but we would be prepared to listen to what was being proposed."
A precedent has already been set with sponsored grounds such as Middlesbrough's Cellnet Riverside stadium and Bolton's new Reebok stadium.Reuse content