Departing party leader Mr Hague has pledged his support for Iain Duncan Smith, because he fears Mr Clarke would cause the party to slide into divisions over Europe.
Mr Clarke said the news came as no surprise and suggested Baroness Thatcher was also likely to back his rival.
Playing down the significance of the backing, he told the BBC's Breakfast with Frost programme that Mr Hague had "one vote like everybody else who has paid their subscription in the Conservative Party".
His supporters say party members might find it hard to value the endorsement of a man who led the party to defeat at the last general election.
The interview came as the two contenders in the Conservative leadership contest prepare to launch their manifestoes.
Ballot papers are due to be sent out on Monday to 330,000 party members as the battle enters its final stages.
Mr Duncan Smith says he is honoured by Mr Hague's endorsement.
Poll
Mr Clarke challenged Mr Duncan Smith to commit himself to keeping Britain in Europe if he becomes leader.
"He has always argued vehemently that we should renegotiate the basis of the treaties and leave if we can't renegotiate them," he said.
As reports suggested that Duncan Smith's leadership manifesto would indicate a desire for closer links with the North American trade zone Nafta, Mr Clarke described his rival's European policy as "a euphemism for leaving the EU".
The Sunday Times and Sunday Telegraph newspapers have also backed Iain Duncan Smith.
The Sunday Telegraph said an ICM poll of 501 people shows Mr Clarke's public popularity is slipping.
The poll also suggests 18% believe Mr Clarke will split the party over Europe, compared to 5% for his rival.
But Mr Clarke said the polls were the most encouraging news he had had as they put him "miles ahead" in questions about who is a natural party leader, a potential prime minister and someone who recognises the need for change.
But former Tory chairman Michael Ancram said Mr Clarke was likely to tear the party apart.
He told Frost: "I'm looking for someone who can reunite the party with exciting, new radical ideas and I think that is Iain Duncan Smith.
"It is important that we have a leader who speaks for the majority of people and on major issues such as Europe and the Euro."
Tory party members have until noon on 11 September to return their ballot papers.
Counting will begin the following day.