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.