The broadcaster said the Prime Minister and Justice Secretary - who are on opposite sides of the campaign - will face questions from a live audience in two special editions of its Question Time programme.
Pro-Brexit Mr Gove will feature in the first of the shows - moderated by usual host David Dimbleby - on June 15 from Nottingham, with Mr Cameron representing the Remain camp on June 19 in Milton Keynes.
The PM has declined to take part in any “blue-on-blue” TV clashes with fellow Conservatives such as Mr Gove or Boris Johnson, arguing he wanted to show the debate went wider than within his own party.
He has already agreed to feature on an ITV show on June 7 straight after Nigel Farage, whose inclusion instead of a figure from the official campaign sparked angry accusations by Vote Leave - firmly denied - of bias by the broadcaster.
Sky News will broadcast separate shows featuring Mr Cameron and Mr Gove on June 2 and 3.
The BBC also announced the line-up for its first debate show to be broadcast on May 26 - which will be hosted in Glasgow by Victoria Derbyshire with an audience of 18 to 29-year-olds.
SNP MP and former Scottish first minister Alex Salmond and the head of the Labour In campaign, former home secretary Alan Johnson, will make the case for continued membership while the anti-EU camps are represented by Tory ex-defence secretary Liam Fox and senior Ukip MEP Diane James.