In a new book by Bob Woodward, the newspaper's investigative reporter, the journalist says that Andrew Card, White House chief of staff, urged Bush to replace Rumsfeld with James Baker, the former US secretary of state, following the 2004 election.
Woodward says Bush decided not to do so after Dick Cheney, the US vice president and political adviser Karl Rove convinced him the move would be seen as an expression of doubt about the direction of the war and expose him to criticism.
The book says Card, with the backing of Laura Bush, the first lady, tried a second time to persuade Bush to fire Rumsfeld around Thanksgiving in 2005. But the president again refused to act.
The comments immediately put the Bush White House on the defensive.
Tony Snow, the White House press secretary, was besieged by questions at his daily briefing and said he had been unable to reach Card, who resigned as chief of staff in March of this year.
The information raises more questions about the administration's handling of the Iraq war only days after a government intelligence report was disclosed suggesting the war was doing more to endanger US security rather than bolster it.
Woodward says President Bush ignored an urgent warning in September 2003 from a top Iraq adviser who said that thousands of extra US troops were desperately needed to end the insurgency in the country.
Robert D Blackwill, Iraq's top adviser on the National Security Council at the time, is said to have issued his warning about the need for more troops in a lengthy memorandum sent to Condoleezza Rice, the US secretary of state.
The book says Blackwill advised that as many as 40,000 soldiers were desperately needed.
State of Denial is Woodward's third book about the administration since the September 11 attacks. It is scheduled for release next week.
While the earlier books were criticised by some as painting Bush as a hero, the current work portrays senior administration officials as being unable to face the consequences of their Iraq policy, the Post said.