Updated: 8:34 p.m.

Senate Majority Leader Harry Reid will begin on Thursday the process for bringing House Republicans’ Cut, Cap and Balance bill to the floor, which could clear the way to move on a package to raise the nation’s debt ceiling as soon as this weekend, the Nevada Democrat announced Wednesday.

Reid intends to act Thursday to set up a procedural vote that would take place Saturday, unless he and Minority Leader Mitch McConnell (R-Ky.) agree to hold the vote sooner. Democratic aides were hopeful that it could be pushed up to Friday, making it possible for the Majority Leader to move Sunday night to set up a procedural vote for a debt ceiling proposal he is negotiating with McConnell.

However, McConnell said on the floor after Reid’s announcement that the chamber should take the two full days available to debate the legislation and that he looked forward to a Saturday vote.

The constantly evolving negotiations to reduce the deficit and raise the debt ceiling, including White House meetings Wednesday, make any schedule tentative.

The Cut, Cap and Balance legislation, which the House passed on a largely partisan 234-190 vote Tuesday night, would implement steep cuts in government spending and set hard discretionary spending caps. It would also make an increase in the debt limit contingent on Congress sending a constitutional balanced budget amendment to the states for ratification.

Though it will not pass in the Democratic-controlled Senate, the vote on the tea-party-influenced measure gives Republicans an opportunity to voice their support for a more conservative plan before having to potentially vote on an alternative package, including the deal that McConnell and Reid are hashing out.

Despite the near impossibility of the Cut, Cap and Balance Act arriving at President Barack Obama’s desk, the White House issued an unusually strong veto threat Monday.

“Neither setting arbitrary spending levels nor amending the Constitution is necessary to restore fiscal responsibility,” the statement of administration policy said. “Increasing the Federal debt limit, which is needed to avoid a Federal government default on its obligations and a severe blow to the economy, should not be conditioned on taking these actions. Instead of pursuing an empty political statement and unrealistic policy goals, it is necessary to move beyond politics as usual and find bipartisan common ground.”

Niels Lesniewski contributed to this report.