WASHINGTON—Efforts to reach a bipartisan deal extending emergency jobless benefits for the long-term unemployed collapsed in the Senate on Tuesday, ending at least for now a push by Democrats to revive a program that started during the financial crisis.

Lawmakers had been working for weeks to restore benefits that expired on Dec. 28, but progress ground to a halt as tensions rose between Republicans and Democrats over the control of Senate votes, a politically sensitive procedural issue that often flares in election years....