The motion to end debate on proceeding to the bill would require 60 votes — that vote is likely to come next week.
Some Republicans have complained that the legislation would provide amnesty for illegal residents before strengthening border enforcement.
Sens. Marco RubioMarco RubioRubio: Turkey attack 'directed' by ISIS Trump: Rivals who don't back me shouldn't be allowed to run for office GOP mega-donor: Trump would cause 'global depression' MORE (R-Fla.), John McCainJohn McCainBush World goes for Clinton, but will a former president? GOP senator: Trump could lose Arizona Senate panel passes bill that would create 4K visas for Afghans MORE (R-Ariz.), Lindsey GrahamLindsey GrahamOvernight Defense: US blames ISIS for Turkey attack | Afghan visas in spending bill | Army rolls up its sleeves Senate panel passes bill that would create 4K visas for Afghans Trump: Rivals who don't back me shouldn't be allowed to run for office MORE (R-S.C.), Jeff FlakeJeff FlakeGOP senator: Trump could lose Arizona McConnell quashes Senate effort on guns Bipartisan gun measure survives test vote MORE (R-Ariz.), Charles SchumerCharles SchumerFormer Gillibrand aide wins NY House primary Senate faces critical vote on Puerto Rico Juan Williams: GOP sounds the sirens over Trump MORE (D-N.Y.), Dick DurbinDick DurbinClinton ally stands between Sanders and chairmanship dream Reid backs House Puerto Rico bill McConnell pledges redo vote on Zika after break MORE (D-Ill.), Bob MenendezRobert MenendezOvernight Finance: Trump threatens NAFTA withdrawal | Senate poised for crucial Puerto Rico vote | Ryan calls for UK trade deal | Senate Dems block Zika funding deal Menendez rails against Puerto Rico bill for 4 hours on floor Overnight Cybersecurity: Senate narrowly rejects expanding FBI surveillance powers MORE (D-N.J.) and Michael BennetMichael BennetCruz-backed candidate wins GOP primary in Colorado Colorado GOP Senate race to unseat Dem incumbent is wide open Ted Cruz chooses sides in Colorado Senate primary MORE (D-Co.) comprise the Gang of Eight.