WASHINGTON (AP) — Senate passage of historic immigration legislation offering citizenship to millions looks near-certain after the bill cleared a key hurdle with votes to spare.

A final vote in the Senate on Thursday or Friday would send the issue to the House, where conservative Republicans in the majority oppose citizenship for anyone living in the country illegally.

Some GOP lawmakers have appealed to House Speaker John Boehner, R-Ohio, not to permit any immigration legislation to come to a vote for fear that whatever its contents, it would open the door to an unpalatable compromise with the Senate. At the same time, the House Judiciary Committee is in the midst of approving a handful of measures related to immigration, action that ordinarily is a prelude to votes in the full House.

Rep. Paul Ryan, R-Wis., said Tuesday that the Senate's advancement of stronger border security measures makes it "even more likely" that immigration reform will pass the House and become law. He said that the House won't take up the Senate bill but will do its own legislation, and added, "the majority of Republicans support the border security" as the keystone of immigration reform. He spoke on CBS' "This Morning."

"Now is the time to do it," President Barack Obama said Monday at the White House before meeting with nine business executives who support a change in immigration laws. "I hope that we can get the strongest possible vote out of the Senate so that we can then move to the House and get this done before the summer break" beginning in early August.

House Minority Leader Nancy Pelosi, D-Calif., said Tuesday she thinks it's important for the House to have its own bill and said, "Let's be optimistic about it."

Pelosi told CNN she thinks it has an excellent chance of passing there because GOP lawmakers are the party's poor showing with Hispanic voters in last year's presidential election "sends an eloquent message" to them.

Obama's prodding came several hours before the Senate voted 67-27 to advance the measure over a procedural hurdle. The tally was seven more than the 60 needed, with 15 Republicans joining Democrats in voting yes.

"I think we're building momentum," said Sen. Bob Corker, R-Tenn., who worked with Sen. John Hoeven, R-N.D., on a $38 billion package of security improvements that helped bring Republicans on board by doubling the number of border patrol agents and calling for hundreds of miles of new fencing along the border with Mexico. Those changes brought border security spending in the bill to $46 billion.

"The bill has been improved dramatically tonight by this vote, there's no question," Corker said. "My sense is we're going to pass an immigration bill out of the United States Senate which will be no doubt historic and I think something that's very, very important to this nation."

Last-minute frustration was evident among opponents. In an unusual slap at members of his own party as well as Democrats, Sen. Ted Cruz, R-Texas, said it appeared that lawmakers on both sides of the political aisle "very much want a fig leaf" on border security to justify a vote for immigration.

Senate officials said some changes were still possible to the bill before it leaves the Senate — alterations that would swell the number of votes in favor.

At the same time, Sen. Roger Wicker, R-Miss., who voted to advance the measure during the day, said he may yet end up opposing it unless he wins changes he is seeking.

Senate Democrats were unified on the vote.

Republicans were anything but on a bill that some party leaders say offers the GOP a chance to show a more welcoming face to Hispanic voters, but which tea party-aligned lawmakers assail as amnesty for those who have violated the law.

At its core, the Senate bill would create a 13-year pathway to citizenship for an estimated 11 million immigrants living illegally in the United States.

The measure also would create a new program for temporary farm laborers to come into the country, and another for lower-skilled workers to emigrate permanently. At the same time, it calls for an expansion of an existing visa program for highly-skilled workers, a gesture to high-tech companies that rely heavily on foreigners.

In addition to border security, the measure phases in a mandatory program for employers to verify the legal status of potential workers, and calls for a separate program to track the comings and goings of foreigners at the nation's seaports and airports.