Both stars have been nominated for Outstanding Drama Performance for their roles as the Time Lord and his assistant Donna Noble in the hit BBC One sci-fi drama.
This year will be the first time male and female stars have been pitted against each other in the acting category, replacing the awards for Most Popular Actor and Most Popular Actress.
David, who has not yet decided whether to return in a fifth series of Doctor Who, has been voted Most Popular Actor for the last two years.
Catherine, 40, and David, 37, will also compete against Ashes to Ashes star Philip Glenister, 45, and Alex Walkinshaw, 34, for his role as Sgt Dale Smith in The Bill.
The awards, hosted by Sir Trevor McDonald, pit Strictly Come Dancing, The X Factor, Britain’s Got Talent and Dancing on Ice against each other in the contest for Most Popular Talent Show.
Ramsay’s Kitchen Nightmares, The Apprentice, Top Gear and This Morning are up against each other for Most Popular Factual Programme.
Jeremy Clarkson and his team have won the trophy for the last two years.
Ant and Dec are heading for their eighth entertainment presenting award, but face competition from Alan Carr, Justin Lee Collins and Paul O’Grady.