The US Senate faces a showdown this week over gun rights, but even supporters of a contentious compromise on background checks conceded Tuesday that they are still short of the necessary votes.

President Barack Obama expressed optimism about the proposed bill and said it would be "unimaginable" if Congress defied strong public support for the measure, especially after the Newtown school massacre in December.

The chamber's top Democrat Harry Reid insisted they had the momentum in the Senate to pass the legislation, which would expand background checks to all commercial firearms sales, including purchases made at gun shows and online.

A deal was being sought that would allow Reid to call for votes on several proposed amendments to the underlying gun bill, beginning with the background check measure as early as Wednesday, he said.

Gabrielle Giffords, the former congresswoman who was shot in the head in 2011 and has emerged as a powerful voice for stronger gun laws, made a dramatic appearance on Capitol Hill to lean on some undecided members.

"I'm optimistic, I think we can get there," her husband Mark Kelly, a retired NASA astronaut, told AFP as he and Giffords, who both strongly back the constitutional right to bear arms, began their last-minute appeals.

But the numbers simply were not there on Tuesday, one of the architects of the compromise conceded.

"We're not ready for a vote," Senator Mark Kirk, who helped craft the legislation with fellow Republican Pat Toomey and Democrats Joe Manchin and Chuck Schumer, told reporters after a debate on the measure officially began.

Supporters would need 60 votes in the 100-member chamber to overcome blocking tactics and ensure final passage. Democrats and independent allies hold 55 seats.

But with a handful of Democrats including Mark Pryor of Arkansas and Mark Begich of Alaska facing tough re-election fights in red-leaning states in 2014, senior Democrat Dick Durbin said his side would need "nine or 10" Republicans on board to meet the threshold.

"They don't have the votes to pass it... and I think they know it," a defiant Iowa Republican Chuck Grassley said in the Senate.

"We haven't voted because despite claims from the other side, background checks are not and never have been the sweet spot of the gun control debate."

Schumer said his side was "trying to get the votes" from moderate Republicans such as John McCain and a handful of Democrats, but such lobbying was being countered by that of the powerful National Rifle Association (NRA).

Schumer and McCain were meeting with Obama later Tuesday to discuss a different subject -- immigration reform -- but the veteran Democratic lawmaker said he had a message for the president on guns: "Just help us pass the bill."

Last week the Senate agreed to debate what has emerged as the most significant guns legislation in nearly 20 years -- 16 Republicans voted to proceed to the bill, arguing that it deserved a debate and a vote, but several have said they would not support it.

Manchin and Toomey spent much of Monday in the chamber seeking last-minute support for the deal, which got a surprise boost when the second-largest US gun rights group, the Citizens Committee for the Right to Keep and Bear Arms, came out in favor of it.

The legislation would expand background checks to all commercial firearm sales, stiffen penalties for gun trafficking and boost school safety measures.

Several amendments are expected to be put forward, including some which gun control advocates say would water down the legislation.

With the bill under threat, Obama has engaged in a full-court press, phoning Republicans as well as Democrats from right-leaning states.