Lord Birt, now the Prime Minister's "blue skies thinker", also said the BBC licence fee could end if there was "insufficient support" from the public.
Speaking at the International Television Festival in Edinburgh, Lord Birt said Radio 4 was a "national treasure" but would benefit from the challenge of a strong rival.
"Radio 4 would perform better if it had competition," he said. "It would be wonderful if there was another Radio 4. There are lots of things about Radio 4 that would have benefited from competition. It is a national treasure ... but I am someone that believes in the benefits of competition."
The BBC offered initial support for the suggestion, saying the corporation would "welcome any competition", but stressed they were Lord Birt's personal views.
Lord Birt, who was director general of the BBC for eight years until 2000, said that if the licence fee were abolished it would "cut off the poorer sections of society". But he warned it may happen eventually.
Andy Duncan, chief executive of Channel 4, backed Lord Birt's comments. "You must have a strong competitor in public service broadcasting, and that must come from Channel 4," he said.Reuse content