WASHINGTON — The bipartisan push to overhaul the nation’s immigration laws took a major step forward Monday evening when the Senate endorsed a proposal to substantially bolster security along the nation’s southern borders as part of a measure that would provide a path to citizenship for 11 million undocumented immigrants already in the country.

The 67-to-27 vote prevented any filibuster of the plan to devote roughly $40 billion over the next decade to border enforcement measures, including nearly doubling the number of border agents to 40,000 and completing 700 miles of fencing. Opponents of the measure questioned whether the security steps would ever be taken and said that the legislation should require that the border be secure before undocumented immigrants could seek legal status.

But the solid bipartisan support for the border security proposal by two Republican senators, Bob Corker of Tennessee and John Hoeven of North Dakota, suggested that advocates of the overhaul had the votes needed to clear remaining hurdles and pass the legislation, which was drafted by a bipartisan “Gang of Eight” senators, perhaps before lawmakers leave town for the July Fourth recess.

Senators and aides said that Monday’s vote offered a preliminary glimpse of how many senators, roughly, would vote for the bill’s final passage. Several senators missed the vote because of flight delays, and Democrats said they could have landed at least 69 votes had all legislators been present.