Richards recently recommended adding 30,000 soldiers to the forces fighting the Taliban, with 5,000 supplied by Britain. The others would come from the United States and the Afghan army.
There are now 8,000 British soldiers in Afghanistan.
"I think militarily there is a case for more troops," Richards told the BBC in a recent interview. "They don't all have to come by any means from the UK."
Richards, 56, joined the army in 1971 and has served in East Timor, Sierra Leone and Northern Ireland.
Dannatt has irritated the government by publicly criticizing the treatment of wounded soldiers and the military pay scale, saying that soldiers get less than traffic wardens. He has also charged that the British presence in Iraq was increasing unrest.