WASHINGTON (AP) — Backers of far-reaching immigration legislation are turning their attention to courting support and counting votes after the Senate pushed the contentious bill over early procedural hurdles.

Two votes to place the bill formally before the Senate and open for amendments each drew more than 80 votes Tuesday, reflecting a bipartisan desire to debate the legislation to remake the nation's immigration laws and open the door to citizenship to millions.

Despite the lopsided votes, Republicans served notice they will seek to toughen the bill's border security provisions and impose tougher terms on those seeking to gain legal status. "This bill has serious flaws," said their party leader, Sen. Mitch McConnell of Kentucky.

At the White House, President Barack Obama insisted the "moment is now" to give the estimated 11 million immigrants living in the United States illegally a chance at citizenship and prodded Congress to send him a bill by fall.

At its core, the bill sets out a 13-year journey to citizenship for the millions of immigrants who arrived in the country illegally through the end of 2011 or who overstayed their visas. The bill also requires a tighter border to prevent future illegal immigration.

Other key provisions would create a new program for low-skilled workers to enter the country and expand the number of visas for high-skilled workers who are particularly in demand in technology firms.

Supporters expressed confidence they could muster the 60 votes needed for the bill to pass the Senate by July Fourth. Democrats control 54 Senate votes, and Republicans 46. But a number of opponents said success was far from assured. And supporters are hoping for closer to 70 votes on final passage to show resounding momentum for the bill and pressure the Republican-led House to act. The safe margin is considered to be 60 votes because that is the number required to fend off a filibuster.

To that end, the bill's four Democratic and four Republican authors were looking for ways to accept Republican amendments on border security and other issues that could win over additional supporters — without making the path to citizenship so onerous that Democratic support is threatened.

"Just because the process has been to date so encouraging does not mean we can take anything for granted. So we welcome constructive input from our colleagues, we want to work with them," said Sen. Chuck Schumer, D-N.Y., one of the authors. "But the one thing none of us will do is condition the path to citizenship on factors that may not ever happen in order to appear tough."

Sen. Marco Rubio, R-Fla., a potential 2016 Republican presidential contender and the author of the bill with the strongest ties to conservatives, said that about half the Senate's Republicans might be prepared to back the measure — but only with stronger border provisions.

An early skirmish took shape over a proposal by Sen. John Cornyn, R-Texas. It would permit the legalization process to begin but require several changes before anyone currently in the country illegally could receive a green card that confers permanent legal residence.

Those changes include apprehension of at least 90 percent of those seeking to cross into the United States at every segment of the southern border, implementation of a biometric exit system at all airports and seaports of entry and a nationwide E-Verify system to check the legal status of prospective employees.

Democratic supporters of the legislation have deemed Cornyn's plan a "poison pill," designed to wreck the bill's chances for passage instead of enhance them. But the Texan told reporters he had some leverage to force changes, if nothing else.

"I think if they had 60 votes to pass a bill out of the Senate, they probably wouldn't be talking to me. And they are," Cornyn said of majority Democrats.