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.

Assuming the Senate can at some point proceed to the bill, Senate Democrats have said the first amendment to be considered would be bipartisan language from Sens. Pat Toomey (R-Pa.) and Joe ManchinJoe ManchinCan Trump rebound after failure on healthcare bill? Path to 60 narrows for Trump pick Overnight Healthcare: Ryan visits White House amid healthcare rubble MORE (D-W.Va.). Their language would expand the current background check requirements only to firearms sales by unlicensed dealers at gun shows, and sales conducted online.

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 ReidHarry ReidAfter healthcare fail, 4 ways to revise conservative playbook Dem senator 'not inclined to filibuster' Gorsuch This obscure Senate rule could let VP Mike Pence fully repeal ObamaCare once and for all MORE (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.