President Bush is not giving up on his plan to legalize millions of unlawful immigrants and will seek to convince skeptical GOP senators that it secures the nation's borders, administration officials said Sunday.

But the Senate's No. 2 Democratic leader said he was uncertain about the prospects for a deal. Sen. Dick Durbin of Illinois cited opposition from a group of Senate conservatives who contend the legislation guarantees amnesty to illegal immigrants.

"We need a breakthrough on the Republican side," Durbin said.

Tony Snow, White House spokesman, and Commerce Secretary Carlos Gutierrez both predicted an immigration bill would pass once Bush meets with lawmakers this week and Congress reconsidered the measure.

Snow said Bush planned to listen to senators' concerns when he meets with GOP senators for a scheduled lunch Tuesday. Bush has been placing phone calls to Republican senators from Air Force One during a European trip.

"I think one of the things you do in this is you listen," said Snow, when asked what Bush planned to say. "And I think one of the concerns a lot of people have, at least around the country, is they say, 'Look, how can we trust you guys to enforce this? You had a border that's been open for 21 years. How can we trust you?"'

"If you take a look at the bill, it is the largest investment ever in border security," he said.

Last week, Senate backers of the immigration bill fell 15 votes short of the 60 needed to limit debate and allow a vote on the measure. Senate Majority Leader Harry Reid, D-Nev., then set the measure aside, calling it "the president's bill" and saying Bush's direct intervention was crucial to reviving it.

The legislation would tighten borders and offer employers more temporary workers from abroad in addition to providing lawful status to an estimated 12 million illegal aliens and putting many of them on a path toward citizenship.

Noting that the measure calls for investing in border security and would hit employers with harsher punishments for hiring illegal immigrants, Snow said the Senate could "wrap this up in two days" if Reid allowed additional debate on it.

Several Senate conservatives have said they were not allowed to offer enough changes to the bill, such as making it easier to detect and deport immigrants who have overstayed their visas or committed other violations.

"Our sense is if Majority Leader Harry Reid brings it back up, which he should, and permits a full debate, ... we're not only going to get a bill, but we're going to get a better bill, and it's going to be one that answers the express objections of a lot of people and, I think, provides a way of answering skeptics on issues like security," Snow said.

Gutierrez agreed. "I believe the votes were there, and I think some senators felt they needed more time and some amendments that did not get their due hearing," he said. "I have no doubt. This is going to go through because it's the right bill."

But Durbin said the Senate already had two weeks of debate on the issue. Reid finally sought to cut off discussion when it became clear that several Republican senators who opposed the proposal offered "amendment after amendment after amendment, to the point where this was bogging down," Durbin said.

"It reached a point, a real impasse," he said. "So it will take some leadership on the Republican side to agree on the number of amendments, to agree that we're not going to waste more time and procedural slowdowns, and really get to the heart of the issue."

Gutierrez said it would be a "big mistake" if senators were to abandon the bill after coming so close to passage.

"I think the big picture is we have a bill here that will strengthen national security, that will improve our economy, that will make us a stronger society and if somebody has a problem with some of the tactics and the clerical issues and the details, let's debate them," he said. "But we should not walk away and keep the status quo."

Snow and Durbin appeared on "FOX News Sunday," and Gutierrez appeard on another cable news network.