Once a front-runner in the Republican field, McCain has fallen behind as his staunch backing for Bush on the unpopular Iraq war and on an overhaul of immigration laws cost him support among both moderates and conservatives.
'Ups and downs'
"We've had ups and downs in other campaigns and we'll have ups and downs with this campaign. I'm very happy with where we are," McCain told areporters at the Capitol, promising to "out-campaign" his rivals.
Nelson was Bush's political director during his 2004 re-election campaign and Weaver was McCain's strategist during his unsuccessful 2000 race for the White House.
"I believe that our military ... is making progress in a number of areas. In other areas they are not. This strategy is the correct one"
John McCain, Republican senator from Arizona
In statements released by the campaign, neither offered a reason for leaving.
"I believe John McCain is the most experienced and prepared candidate to represent the Republican Party and defeat the Democratic nominee next year," Nelson said.
Rick Davis, McCain's campaign manager in 2000 and chief executive officer of this campaign, will take over managing McCain's White House run.
The shake-up follows a campaign reorganisation and cutbacks in staff last week, the result of a weak fund-raising quarter that left McCain with just $2 million in the bank.
McCain has been lagging behind Rudy Giuliani, former New York Mayor, in polls, and trails Giuliani and Mitt Romney, former Massachusetts governor, in raising money for the November 2008 election.
McCain's disappointing take of $11.2 million in the last three months, and his $2 million in the bank, prompted a flood of criticism and questions last week about spending and strategy within the campaign.
A Republican strategist said it would be difficult for McCain's campaign to pull out of its death spiral.
"They can decide to crank up the bus and try to recapture the magic from 2000, but once it's gone it is hard to get back," said the consultant, who asked not to be named. "The fund-raising community just closes their cheque books."
Support for Bush
Tuesday's announcement came as McCain, who visited Iraq last week, reiterated his support for Bush's new war strategy and the Senate reopened debate on the issue.
While Democrats are pressing for a plan to withdraw troops and several Republicans have defected from Bush over the war, McCain said it would be a mistake to abandon Bush's strategy now.
"I believe that our military in cooperation with Iraqi security forces is making progress in a number of areas. In other areas they are not," McCain said. "This strategy is the correct one."