The effort to reform the nation’s immigration laws took another important step forward Monday when a Republican proposal to bolster security along the U.S.-Mexico border cleared a key procedural hurdle by a margin that bodes well for its eventual approval in the Senate.

The Senate voted 67 to 27 to proceed to debate on the proposal, exceeding the threshold necessary to move forward but falling short of the 70 votes that some supporters had hoped it might earn. The vote was one of just a final few steps left before the Senate is expected to give final approval to the bipartisan measure later this week.

Fifteen Republicans voted with 52 members of the Senate’s Democratic caucus in support of the plan, which calls for doubling the size of the U.S. Border Patrol to about 40,000 agents, the completion of 700 miles of fencing along the southern border and the expanded use of radar and unmanned aerial drones.

The deal was designed to garner the support of a collection of Republicans including Lamar Alexander (Tenn.), Kelly Ayotte (N.H.), Orrin G. Hatch (Utah) and Roger Wicker (Miss.).

But a block of conservative Republicans stood firm against the plan, including Senate Minority Leader Mitch McConnell (R-Ky.) and his deputy, Sen. John Cornyn (R-Tex.), who had pushed for a stronger border security plan that was rejected last week.

Voting on the motion to proceed was kept open Monday evening because flights carrying several senators back to Washington were delayed by severe weather and issues with an airplane that veered off a runway at Reagan National Airport. Despite the longer voting time, six senators still missed the vote.

Sen. Charles E. Schumer (D-N.Y.), a key Democratic supporter of the immigration bill, said Monday’s vote “shows that the pro-immigration forces on both sides of the aisle continue to make progress. We realize we have a long hard road ahead of us, but this vote puts the wind at our back.”

In voting no, several Republicans complained Monday that the amendment doesn’t go far enough to bolster border security and that Senate Democrats rushed consideration of the proposal while blocking votes on other GOP proposals.

“What is the rush? Why are we proceeding gangbusters?” said Sen. Ted Cruz (R-Tex.), a vocal critic of the process. He charged that colleagues of both parties were seeking “a fig leaf” that would permit them to say they bolstered security along the border, “when in fact this bill does not.”

McConnell agreed, saying that there “is simply no reason we need to end this debate now in order to meet some artificial deadline determined by the majority leader’s summer schedule.”

It has long been the goal of Senate Majority Leader Harry M. Reid (D-Nev.) and the bipartisan “Gang of Eight” that wrote the immigration bill to hold a final vote before the start of the July Fourth congressional recess. Their hope is that the House will devote most of July to consideration of immigration proposals and approve a bill before the month-long August recess, which would allow negotiators to spend the last few weeks of the summer working on a final deal that Congress could approve in the fall.

But consideration of the immigration bill in the House remains in doubt after lawmakers of both parties rejected a broad five-year farm bill that revived concerns about whether House Speaker John A. Boehner (R-Ohio) can cobble together enough support among Republicans and Democrats to pass significant legislation.

Boehner has said that he will only hold a vote on an immigration measure that enjoys support from a majority of Republicans but that he also hopes to proceed in a bipartisan fashion. But GOP aides have said that cooperation with Democrats may prove impossible after they withdrew support for the farm bill.

In a notable sign of conservative support, Arizona Gov. Jan Brewer (R) said Monday that she supports the border security plan that advanced Monday.

“I think that what we’re seeing taking place in the Senate is a victory for Arizona,” she told Fox News. “I’m glad that they finally decided to talk about the ‘border surge’ that we’ve called out for since 2010.”

Brewer later clarified via Twitter that her support applied only to the border security amendment and not the overall bill. She has emerged as a conservative champion in recent years for critiquing the Obama administration’s handling of border security.

At the White House, President Obama sought to rally business leaders in support of the measure by meeting with a group of corporate and small-business executives.

“It’s not a bill that represents everything that I would like to see; it represents a compromise,” Obama told reporters before the meeting.

“If we get this done — when we get this done — I think every business leader here feels confident that they’ll be in a stronger position to continue to innovate, to continue to invest, to continue to create jobs, and ensure that this continues to be the land of opportunity for generations to come,” he added.