A successful vote Thursday morning will bode well on the actual vote to proceed to the bill, which could take place quickly if senators vote to end debate. But if Republican senators want to stall the bill, the vote to proceed would be pushed to Friday afternoon.
It also prohibits the government from maintaining a national firearms registry, and calls for stiff penalties for entities who keep or misuse background check records.
In its current form, S. 649 requires universal background checks for all gun sales.
Senate Majority Leader Harry Reid (D-Nev.) said the Senate would not finish the bill this week, and said various other amendments to the bill are likely along the way.
In the House, members will start legislative work at noon on legislation that would freeze the work of the National Labor Relations Board (NLRB).
The bill is meant to keep the NLRB from issuing new labor management decisions, or enforcing decisions reached since January 2012, until there is a final determination of whether President Obama appointed members to the NLRB constitutionally.
Last year, Obama "recess appointed" three people to the NLRB, two of whom are still there: Sharon Block and Richard Griffin. But a federal court found that the appointments were not constitutional, as the Senate was not in recess at the time.
The Preventing Greater Uncertainty in Labor-Management Relations Act, H.R. 1120, would freeze the board's work until a Supreme Court ruling that decides the question of the legitimacy of the appointments.
The House is expected to approve a rule on Thursday that governs floor debate, and then pass the bill itself on Friday. The rule allows no amendments.