According to a report in the Sunday Times, GCap has been in talks with the 37-year-old for several weeks.
Brady has run Birmingham City since 1992 when she was promoted to managing director by the club's co-owner, porn baron David Sullivan, who also owns the Daily Sport.
She has the nickname "the first lady of football" and is only woman to run a top-flight club.
The move will come as shock to her club as it struggles to win promotion back to the Premiership after being relegated at the end of last season.
It is not Brady's first association with radio. In 2003, she was appointed as non-executive chairman of Emap's Kerrang! Radio bid for the West Midlands radio licence, which it successfully won.
If appointed by GCap, she would replace Keith Pringle who was axed as managing director in April 2006 along with a number of other directors, including programme director Nik Goodman.
The pair were replaced by Scott Muller, who joined as programme director from the Nova radio station in Australia, in an effort to reverse Capital's fortunes.
In August, Capital Radio's breakfast show hosted by Johnny Vaughan lost 111,000 listeners. In a bad second quarter for Capital, its breakfast show pulled in just 852,000 listeners, compared with 963,000 in the previous quarter and 1.08m a year ago. The figures represented a new low for the station and fopr Vaughan, who inherited a 1.38m breakfast audience from Chris Tarrant in the spring of 2004.
With Brady in as managing director, Muller would focus on his strategy of fewer ads. Brady who survived a brain aneurysm earlier this year is married to Derby County footballer, Paul Peschisolido.
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