Jeff Stelling announced last week that he would be exiting the long-running Channel 4 daytime quiz at the end of the year.
Channel 4 is currently searching for a replacement to join Rachel Riley, and impressionist Bremner and Countdown regular Brandreth are currently being backed by punters with odds of 9/2 and 5/1 respectively.
Bremner turned down an offer to host the show after Des O'Connor quit in 2008, but admitted that it was a tough decision and didn't rule himself out of taking the role in the future.
Other potential new hosts include Radio 2's Aled Jones (5/1), Lisa Tarbuck (5/1), Michael Aspel (8/1), Tom O'Connor (8/1), Terry Wogan (10/1) and Anne Robinson (10/1).
Countdown airs weekdays on Channel 4.
> Sir Terry Wogan in line to host 'Countdown'?
> More TV news