The mounting controversy centres on his leadership of US military efforts in Iraq.
"I have no confidence in Rumsfeld," said US Senator Chuck Hagel, who became the latest in a steady stream of leading Republicans to express a lack of confidence in the Pentagon's top civilian official.
Hagel called the US military's continued problems "a manifestation of a clear lack of cogent, clear, straight-talking planning in a post-Saddam Iraq".
"We're in there now 21 months. Things are worse than they've ever been," said Hagel.
Rumsfeld has come in for a barrage of fresh criticism after admitting that he had not personally signed Pentagon condolence letters to families of soldiers killed in Iraq. But he has vowed to do so in the future, according to a Washington Post report on Sunday.
"I have no confidence in Rumsfeld"
"I wrote and approved the now more than 1000 letters sent to family members and next of kin of each of the servicemen and women killed in military action," Rumsfeld said in a statement to the military newspaper Stars and Stripes.
"While I have not individually signed each one, in the interest of ensuring expeditious contact with grieving family members, I have directed that in the future I sign each letter," the defence chief said, according to the Post.
An outcry ensued after the Stars and Stripes reported in late November that the Pentagon was using a device to stamp Rumsfeld's signature on the letters, and quoted recipients who said they were insulted.
In recent days, several top Republicans, including senator and former presidential candidate John McCain, have criticised Rumsfeld, whom Bush has asked to stay on amid a cabinet re-shuffle following his November election victory.
Senator McCain says he does not
fancy working with Rumsfeld
Rumsfeld has drawn fire since seeming to dismiss the concerns of a US soldier at a so-called town hall meeting in Kuwait who asked him about the lack of armour for US military vehicles in Iraq.
In widely broadcast and published remarks, Rumsfeld responded: "You go to war with the army you have, not the army you might want or wish to have at a later time."
Numerous Democrats in the US legislature have called on Rumsfeld to step aside, including Rhode Island Senator Jack Reed, who told CBS on Sunday: "I believe that the country will be better served by Secretary Rumsfeld's departure."
Some support left
But other top Republicans offered qualified support to the beleaguered defence secretary.
"He should be held accountable, and he should stay in office," said Richard Lugar, chairman of the Senate Foreign Relations Committee, speaking on NBC television's Meet the Press programme.
"He needs, at this point, to listen, and he is listening."
"Secretary Rumsfeld is doing a spectacular job, and the president has great confidence in him. He is helping to lead our troops as they meet an awesome responsibility to protect us and to bring freedom to others"
White House chief of staff
Meanwhile, the head of the Senate's Armed Services Committee - without giving Rumsfeld a vote of confidence as such - said changing the Pentagon leadership in the middle of the Iraq conflict would be imprudent.
"We should not at this point in time entertain any idea of changing those responsibilities in the Pentagon," Senator John Warner said on NBC, adding that "we're going to have a tough period after" Iraq's 30 January elections.
For its part, the Bush administration, which repeatedly has voiced unqualified praise for the 72-year-old Rumsfeld, did so again on Sunday, with White House chief of staff Andrew Card calling the defence secretary's performance "spectacular".
"Secretary Rumsfeld is doing a spectacular job, and the president has great confidence in him. He is helping to lead our troops as they meet an awesome responsibility to protect us and to bring freedom to others," Card said.
"He's also transforming the military. And any time you do that, there are controversies."