WASHINGTON (Reuters) - About $46 billion in border security was added on Wednesday to a comprehensive immigration bill headed toward bipartisan passage in the U.S. Senate, but Republicans in the House of Representatives voiced strong opposition. The Democratic-led Senate is expected to approve the White House-backed bill on Thursday or Friday and send it to the House, which likely will not even bring it up for a vote, largely because of resistance to a proposed pathway to citizenship for up to 11 million undocumented immigrants in the United States. "I think this issue is getting ready to slow down a lot," Republican Representative Tom Cole of Oklahoma told Reuters. Cole predicted that Republican leaders will become preoccupied with an approaching debt limit problem and the September 30 deadline for providing new funds to keep the government running. At the end of 2012, Cole was a harbinger of what was to come in a "fiscal cliff" battle when he became the first House Republican to publicly back a tax increase on the wealthy that President Barack Obama was pushing. The increase ultimately became law with some Republican support. "Until you get the fiscal issues settled, I wouldn't be looking for any big immigration fight if I was in leadership," said Cole, a deputy House Republican whip who helps gauge support within party ranks for various bills. In the Senate, except for the final vote on passage, the battle over the bipartisan bill is essentially over. The bill would give legal status to most of the 11 million undocumented foreigners in the United States, allowing them to live and work here without fear of deportation. The first major rewrite of U.S. immigration laws since 1986, it would also provide these illegal immigrants with a chance to gain U.S. citizenship by 2026. In a show of bipartisan Senate support, 69 of the chamber's 100 members, including 15 of 46 Republicans, voted for the border security amendment that was constructed to lure more Republican backing for the overall bill. It would spend an additional $46 billion over 10 years, far beyond the $6.5 billion initially envisioned by the authors of the legislation, to double the number of agents on the U.S.-Mexican border to about 40,000. It would also authorize money to complete a 700-mile (1,100-km) fence on portions of the border and purchase high-tech surveillance equipment to detect illegal crossings. Democrats dodged a potentially divisive fight over immigration policy and same-sex marriage when the U.S. Supreme Court struck down a law defining marriage as the union of a man and a woman. Without that ruling, Senate Judiciary Committee Chairman Patrick Leahy might have sought a vote on his amendment allowing citizens to petition for their foreign-born, same-sex spouses to immigrate to the United States. Despite all the progress in the Senate on Wednesday, members of Congress and immigration advocacy groups pushing the legislation had reason to be only cautiously hopeful. Senate Republican leaders remained steadfastly opposed to the measure, somewhat dampening its prospects, and some House Republicans were outright disdainful of the Senate's work. While some House Republicans have been working with House Democrats on a comprehensive immigration bill - one that contains a pathway to citizenship for the 11 million - many in the party want only a limited immigration bill to advance this year, which Democrats would oppose. "Our viewpoint is that we pass border security first so if anything comes to a vote in the House, it'll be that," said Representative John Fleming of Louisiana, a Tea Party activist. Speaking to reporters following a closed-door meeting of Republican members and their leaders, Fleming said House Speaker John Boehner was adamant that the House would not simply take up the Senate's bill in July. "He said in every way, shape and form this is not happening," Fleming said. Asked about a bipartisan House group that is thought to be crafting legislation similar to the Senate bill, Fleming said, "They're an apparition ... I don't even know who they are so how can we take them seriously. Republican Representative Raul Labrador of Idaho, who until recently was part of the House bipartisan group, told reporters, "The American people are not clamoring for a path to citizenship," adding that they instead want illegal immigration stopped at the border. It is only the activists that are clamoring for a path to citizenship and you guys in the media keep ... being shills for the activists. Some Senate Republicans continued to hammer away at the $46 billion border security amendment that has helped turn the tide in the bill's favor. Republican Senator Charles Grassley of Iowa voiced skepticism that the 20,000 additional border security agents actually would be hired over the next 10 years. Additional reporting by Rachelle Younglai, Editing by Vicki Allen, Cynthia Osterman and Mohammad Zargham