Men pass a mural in Altar, Mexico, honoring migrants who have died trying to reach the United States. The Senate will resume work on an immigration bill in June.

Washington ? Lawmakers are bracing for political blowback from liberal and conservative critics of a broad immigration reform measure that’s on hold as they head home for a weeklong break.

Supporters and opponents said they would use the time off to influence senators who will face major immigration votes when they return to complete the bipartisan measure. The bill would legalize millions of unlawful immigrants, tighten border security and mandate that employers verify they are hiring legal workers.

“As we head into the Memorial Day weekend, members of the Immigration Reform Caucus will go back to their districts and remind their constituents about the long-lasting damage passing an amnesty bill would have on our efforts to address illegal immigration,” said Rep. Brian Bilbray, R-Calif., a vocal opponent of the measure.

A coalition of organizations actively pressing for passage of the bill – including labor, Hispanic advocacy and religious groups – also plan a strong push to encourage lawmakers who support it not to back down.

“They’re going to be targeted by a small but loud anti-reform minority,” said Clarissa Martinez of the Coalition for Comprehensive Immigration Reform. “They must not buckle under the pressure.”

Lawmakers already are hearing vocal opposition to the measure.

“I have learned some new words from some of my constituents,” said Sen. Jon Kyl, R-Ariz., an influential conservative who is under intense criticism in his border state for crafting a bipartisan deal to create a guest worker program and a path to citizenship while tightening border security.

The Senate began considering the bill this week and will return to it in June.

Give-and-take

The measure includes conservative-backed initiatives such as the worker verification program to prevent illegal immigrants from getting jobs, and a new point system to prioritize skills and education over family in deciding who can immigrate in the future.

Liberals decry the point scheme as unfair to families and are vehemently opposed to a guest worker program that would allow laborers to come to the U.S. for temporary stints without a guarantee they would be able to stay and eventually gain citizenship.

But it also includes a long-sought liberal priority: granting legal status to the nation’s estimated 12 million illegal immigrants. Conservatives view that as an unacceptable amnesty program.

President Bush, who has made enactment of an immigration measure a top domestic priority, said he wasn’t surprised at the tenor of the debate, and he urged lawmakers to look beyond it and deliver him a bill anyway.

“We’ve been through immigration debates in this country, and they can bring out the worst sometimes in people,” Bush said. “But the question is, will members of Congress rise above politics?”

Backers of the deal said they were confident about facing their constituents during the break.

“I look forward to going back home and standing up for what I believe to be the right answer, listening to criticism. But time is on our side because our product is better than those who want to do nothing,” said Sen. Lindsey Graham, R-S.C.

Some lawmakers were uneasy, however, about trying to explain the complex bill to voters.

Sen. Mitch McConnell, R-Ky., the minority leader, said opposition from the political extremes is “pretty obvious from the phone calls that all of us are getting.”

However, he added, “the American people would like us to do something.”

Poll finds support

There’s wide support for the changes proposed in the immigration legislation, according to a New York Times-CBS News poll conducted May 18-23 and released Friday.

Two-thirds of the 1,125 adults polled said they supported letting illegal immigrants with good job records and no criminal histories win legal status. That was supported by roughly equal proportions of Republicans, Democrats and independents.

Another two-thirds also said they supported creating a guest worker program for future immigrants. Of those who favored it, more than half said the guest workers ultimately should be allowed to apply for citizenship, if their job records are strong and they don’t commit crimes.

Three-fourths supported stronger penalties against those who employ illegal workers.