4/15/13 3:18 PM EDT Updated: 4/16/13 10:00 AM EDT

The White House and Senate Majority Leader Harry Reid are shy of the 60 votes they need to move the bipartisan compromise bill on background checks for gun sales.

Vice President Joe Biden has been personally calling senators to urge them to support the measure, Democratic aides say.

Majority Whip Dick Durbin (D-Ill.) began a whip count on Monday, Democratic aides said, and Biden has been pressuring his fellow Democrats to fall in line. Biden’s office did not return an email seeking comment.

Sens. Joe Manchin (D-W.Va.) and Pat Toomey (R-Pa.) are co-sponsoring the proposal that would expand background checks for commercial gun purchases — including those at gun shows and online — opening a pathway for the biggest change in U.S. gun laws in nearly two decades.

A cloture vote on the Manchin-Toomey could come by Thursday, Democrats said.

Toomey on Monday afternoon acknowledged they don’t yet have the votes.

“I’m cautiously optimistic. We’re not there at the moment but were working on it,” Toomey said.

Manchin also said supporters were short of the votes needed to overcome a GOP filibuster, but he remained hopeful that it would pass.

“We’re still working it, it’s very close,” Manchin told reporters. “I agree [with Toomey], it’s very close.”

Manchin added: “I’m talking to everybody.”

Manchin will appear with former Rep. Gabrielle Giffords (D-Ariz.) on Tuesday to press for its passage. Giffords was seriously wounded in a January 2011 shooting that left six others dead. Giffords and her husband, former astronaut Mark Kelly, have become high-profile backers of new gun regulations.

Reid and Minority Leader Mitch McConnell (R-Ky.) are attempting to hash out a floor procedure laying out votes on the Manchin-Toomey proposal.

If Democrats cannot overcome a GOP filibuster on the background checks bill, any chances for major gun control legislation being enacted in the wake of the Newtown, Conn., shooting would diminish dramatically.

With Republicans filibustering the Manchin-Toomey proposal, a cloture vote on the bill is likely to take place on Thursday at the earliest.

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