WASHINGTON — The Senate began a drive yesterday to revive legislation to overhaul immigration, with the Bush administration predicting enough support to at least overcome a procedural hurdle that could short-circuit the effort.

“We’ve got a couple of days of hard work ahead of us,” President Bush said yesterday at a White House briefing for supporters of the bill, the centerpiece of his domestic agenda. The legislation “addresses the needs of a failed system,” he said.

The Senate must first overcome a potential stumbling block by voting to permit debate to resume on the measure. That will require bipartisan support to produce 60 votes, or three-fifths of the Senate.

“We’re optimistic that the 60 votes will be there,” Joel Kaplan, Bush’s deputy chief of staff for policy, told reporters on Monday.

After the legislation was derailed this month in the Democratic-controlled Senate, supporters negotiated a deal to rescue it by allowing votes on two-dozen amendments. A number of Mr. Bush’s fellow Republicans have labeled the package amnesty for lawbreakers and are backing changes aimed at stopping illegal border crossings from Mexico.

The measure would offer 12 million illegal immigrants a path to citizenship, create a guest-worker program to help American employers fill low-paying jobs and increase penalties for companies that hire illegal immigrants.

The outcome of the preliminary vote to resume debate will depend in large measure on the ability of the Republican leader, Senator McConnell of Kentucky, to deliver enough support to help the majority leader, Senator Reid, a Nevada Democrat, reach 60 votes. Mr. McConnell has said he won’t decide whether to vote for the measure until votes are completed on the amendments.

“There are enough undecided votes at this point to stop the bill,” Senator DeMint, a Republican of South Carolina, told reporters yesterday. But he said, “when you’re going against the president” and Senate leaders in both parties, “there’s a lot of arm twisting.”

Mr. Kaplan, in a conference call with reporters, said many Americans have “legitimate skepticism” about the government’s commitment to securing the border given the results of the last major revision of immigration in 1986.

Mr. Bush has endorsed including in the legislation $4.4 billion to improve border security to ease some of those concerns.

Mr. Kaplan said a “big concern” is an amendment to delete the legislation’s requirement that employers check their entire workforce to identify any undocumented aliens.

White House officials are urging senators to keep intact provisions that require such checks along with verifying that all new hires have legal status, Mr. Kaplan said.

Senator Murkowski, a Republican of Alaska, said she would vote to resume debate. “What we need to do is get to the bill,” she said. “I am ready to go.”

Mr. Murkowski declined to say whether she would vote to approve the measure until the amendment process is completed.

The key test for the legislation probably will come in two days when Reid seeks to shut off debate, which also takes 60 votes.

Senator Sessions of Alabama told reporters “some people could justify” supporting the debate’s resumption today yet still vote to block final passage on grounds the legislation “won’t work.”

In a floor speech, he said the procedure agreed to by both parties will let Mr. Reid “completely control the amendment process and bring the bill up to a final vote with amendments only he has approved.”

Mr. Sessions and other opponents pleaded with their Republican colleagues to oppose resumption of debate, saying a vote to resurrect the measure was tantamount to passing it.

“Any senator who votes” to resume debate “is voting to pass this final bill,” Mr. DeMint said.

The legislation stalled June 7 after Senate leaders fell 15 votes short of the 60 needed to conclude debate.

Seven Republicans joined 37 Democrats and Senator Lieberman, a Connecticut independent, to support curtailing debate. Thirty-eight Republicans, 11 Democrats, and Vermont independent Bernie Sanders opposed the debate’s conclusion.

Senator Durbin of Illinois, the Senate’s no. 2 Democrat, told reporters last week that at least 24 Republican votes will be needed to shut off debate.