A hearing was scheduled for Monday on the Bush request.
Bush's chief representative in Florida, James A. Baker III, announced the attempt to secure a federal injunction to prevent the hand counts. He said the hand count raised a "potential for mischief" not present when automated machines tally ballots.
The move came a day after the Republicans criticized the Democrats for threatening to take the ballot issue to court, a step the GOP said would lead to endless wrangling in a number of states.
However, Baker blamed Democrats for starting the legal battle because their supporters - though not the Gore campaign itself - had already filed lawsuits in Florida.
Gore's campaign chairman, William Daley, rejected suggestions that the vice president's campaign was hoping to win on a "legal technicality." Speaking at a news conference later Saturday in Washington, Daley said he wanted to make sure "you won because you won more votes than the other guy."
Appearing with Daley, Gore's representative in Florida and former Secretary of State Warren Christopher, said, "We call on the Bush campaign to withdraw its suit." Signaling that Gore was prepared to extend the critical dispute even further, he added: "We've been considering various other options. No decision has been reached on that."
It was unclear what those other options might be.
The legal battle was most intense here in Palm Beach County, a Democratic stronghold and a major source of support for Gore. After hours of delay, election workers began simultaneous manual and mechanical recounts shortly after 2 p.m. EDT. They said the counts would take hours.
Workers brought in silver metal boxes from four precincts, broke the seals and took the ballots out. Six teams of three counters and two observers peered closely at each ballot to determine who was voted for. Then they placed each ballot into separate piles.
Palm Beach County Judge Charles Burton, a member of the county's canvassing board, said despite the request from Bush for an injunction, the recount would proceed as planned. .
``All of you can go file for an injunction, but until we get a court order it doesn't mean anything,'' he said. ``If we get one we'll read it and we will abide by it.''
The hand recount was supposed to take place in four precincts - one in Palm Beach Gardens, two in Boca Raton and one in Delray Beach. Originally, three counties were chosen, but another was added Saturday because the first three didn't add up to 1 percent of the vote, as required by state law. The ballots arrived at the government center under police escort.
Elsewhere in Florida on Saturday:
About 26,000 votes in Duval County were disqualified and never counted when voters punched more than one candidate on their ballot or failed to vote for president. The county is solidly Republican.
Bush's campaign argued in its application for the federal injunction that there was a need to ``preserve the integrity, equality, and finality'' of the vote.
Baker said the hand-count raises ``the potential for mischief'' and was less accurate than the automated machine recount that already has been completed.
Democrats wanted a recount by hand so that ballots can be examined more closely. If there is a change in the sample count, county officials will then decide whether to do a recount by hand of the entire county.
Barry Richard, a lawyer for the Bush campaign, said one reason they are challenging the manual recount is that by choosing selective precincts, the recount is not giving equal access to people.
``You can keep counting and counting forever,'' he said. ``When do you stop?''
Democrats said thousands of votes in Palm Beach County and elsewhere in Florida may not have been counted because the tiny piece of paper punched out for a candidate did not completely dislodge. About 30,000 ballots were rejected in Palm Beach County alone because they had two or more holes punched for president - or computers didn't detect any holes at all.
Palm Beach County Democrats also complain that the county's ballot was so confusing that many Gore voters mistakenly voted for Reform Party candidate Pat Buchanan. Florida election officials said Friday the ballot did not violate state law, as several lawsuits filed by Democrats contend.
In response to one of those lawsuits, a circuit judge on Thursday issued a preliminary injunction barring the county's canvassing commission from certifying the final recount results until a hearing Tuesday.
Meanwhile, officials said 6,686 ballots were not counted in Broward County because the computer did not recognize any selection. Broward election officials voted 2-1 to do a hand-recount of three precincts beginning Monday. If there is a change, they also will consider a full hand-recount.
On Friday, Secretary of State Katherine Harris said Bush had 2,910,074 votes to Gore's 2,909,114, a difference of 960, with one county still to be recounted - Palm Beach County where the AP showed a big gain for Gore.
The totals from the AP canvass were Bush 2,910,198, Gore 2,909,871.
Meanwhile, polls showed that Americans by a 3-to-1 margin say it's more important to make certain the vote count in Florida has been fair and accurate than to resolve matters quickly.
Just over half oppose waiting beyond the wrap-up of the Florida recount, including absentee votes, due by Friday.
A Newsweek poll out Saturday said two-thirds of Americans think Al Gore did the right thing when he withdraw his concession in the very close presidential election that has not yet been decided.
In a CNN-Time poll released Saturday, four in 10 said Gore should wait for any court decisions on contested ballots in Florida. A fourth said Gore should concede immediately, while another fourth thought he should concede if he loses the official vote that wraps up next Friday. More than half of those surveyed, 54 percent, said in the CNN-Time poll that Bush should not ask for recounts in other states, which the Bush campaign has hinted it will do in Iowa, Wisconsin and New Mexico, where Gore came out ahead. People were evenly split on the question of a new election in Palm Beach County, where there have been widespread complaints about voter confusion over an unorthodox ballot style.
But two-thirds were opposed to a new election in Florida and three-fourths opposed to a new election nationwide.
Just over half of Americans in the CNN-Time poll said the Gore campaign is acting responsibly while the votes are being recounted in Florida. A third said his campaign is acting irresponsibly.
Seven in 10 said the delay in knowing who is president and the uncertainty about the vote are a sign of strength in the nation's political system, while 24 percent said it was a sign of weakness.
When people were asked if the electoral college currently used to select a president, six in 10 said it should be replaced by selection with the popular vote. The electoral college gives each state a certain number of votes in line with their congressional representation. Gore currently leads the popular vote nationally, so if Bush wins the election he would have done so without winning the most votes.
The Time-CNN poll was conducted Friday with 1,154 adults and had a 3 percentage point error margin. The Newsweek poll was conducted with 1,000 adults Thursday and Friday and has an error margin of 4 percentage points.