In a letter to parliamentary colleagues on Wednesday, in which they called for the secret ballot, Hoon and Hewitt said: "This is a clear opportunity to finally lay this matter to rest.
"The continued speculation and uncertainty is allowing our opponents to portray us as dispirited and disunited."
David Cameron, leader of the opposition Conservative Party, said the move showed Brown's government was "completely divided".
"We cannot go on like this, we've got to have an election and a change of government," he told BBC radio on Thursday.
The poll in the Sun newspaper showed Labour trailing the Conservatives by nine percentage points, and UK media noted that a slow and tepid response to the ballot call by many of Brown's most senior colleagues may cause longer-lasting damage.
The survey, carried out on January 5 to 6, showed the Conservatives with 40 per cent support, Labour on 31 per cent and the Liberal Democrats on 17 per cent.
Almost three-fifths of voters, or 58 per cent, said a new Labour leader would not affect their decision.
The timing of the plot was a surprise as Labour had started to recover some ground from the Conservatives in recent polls.
Peter Mandelson, Brown's business minister, dismissed the ballot call as nothing more than a distraction.
"I do not have a queue of cabinet ministers at my door or on the phone saying they want to change the leader," he said.
"I didn't have to arm twist or persuade anyone [to endorse Brown]."
|Miliband took over six hours to pledge his support for the prime minister [Reuters]
David Miliband, the country's foreign minister and a leading candidate to replace Brown, took over six hours to pledge his support, while other ministers gave their backing with varying levels of enthusiasm, newspapers noted.
The BBC reported that those behind the ballot call thought they had the support of six senior ministers.
The broadcaster's political editor named the potential cabinet rebels as Miliband, Harriet Harman, Bob Ainsworth, Jack Straw, Jim Murphy and Douglas Alexander.
"The plotters may have exposed their own weakness and may soon look like a spent force," Britain's Guardian newspaper said.
"But Brown too has been wounded and weakened once again, just when he and Labour needed to gather their strength to fight the enemy beyond."
Brown's critics say he lacks charisma and his ratings have suffered during a deep recession and over his handling of the increasingly bloody campaign in Afghanistan.