Brydon's appointment is understood to be awaiting approval from business secretary Lord Mandelson.
It is possible that Brydon, who chaired market research group Taylor Nelson Sofres until it was taken over by WPP last year, will not accept the role as he is said to be in discussions with several companies about different roles.
If he does take the job Brydon will be expected to oversee the potential sale of a minority stake in Royal Mail as recommended by Richard Hooper, the former deputy chairman of Ofcom, in a report last month.
Private equity group CVC Capital Partners, which already owns stakes in postal services in Belgium, Denmark and Sweden, and Dutch postal group TNT have already expressed interest in making offers.
The decision to part-privatise Royal Mail has angered trade unions and Labour MPs with more than 60 backing an Parliamentary motion claiming the moves "risk fracturing one of Britain's greatest public services".
Leighton is due to stand down as Royal Mail chairman in March after seven years in the job. During his time at the company the group's operations have been restructured and working practices modernised but not without controversy, including the closure of thousands of Post Office branches.