The 72-year-old announced on Wednesday that he was quitting the show after this year’s series.
According to The Sun, Anton believes he has the backing of Len himself and former presenter Sir Bruce Forsyth for the high profile job.
But Brendan, who is a judge on the New Zealand version of the show, thinks he has more relevant experience.
As it stands at the moment, Anton is the early favourite with Ladbrokes making him 2/1 favourite to take a seat next to Bruno Tonioli, Craig Revel Horwood and Darcey Bussell on the judging panel. Brendan has odds of 20/1.
Other ‘Strictly’ stars thought to be in the running include former pros Karen Hardy, Ian Waite and Erin Boag as well as former judge Arlene Phillips, former winner Tom Chambers and ‘It Takes Two’ presenter Zoe Ball.
Explaining his decision to quit, Len said: “In 2004, I was asked to take part in a brand new BBC Saturday night show and who would have thought me, old Len Goodman, would still be part of this amazing series more than 10 years on.
“This adventure began when I was 60 and now that I’ve reached my 70s, I’ve decided after this year it’s time to hand the role of Head Judge to someone else.
“It is an honour being part of the wonderful ‘Strictly Come Dancing’ family and I’m looking forward to my last series very much and to whatever comes next.”