Washington (CNN) A bipartisan bill to let families victimized by the 9/11 terrorist attacks sue Saudi Arabia ran into sharp setbacks Monday, as the White House threatened a veto and a GOP senator privately sought to block the measure.

The move comes as presidential candidates from both parties are seizing on the legislation to score points with New York voters ahead of Tuesday's critical primary there.

And it has pit the likely next Senate Democratic leader, Chuck Schumer of New York, squarely against the Obama administration.

The White House and State Department are bluntly warning lawmakers not to proceed with the legislation over fears it could have dramatic ramifications for the United States and citizens living abroad to retaliatory lawsuits. The President lands in Riyadh Wednesday for talks with Saudi Arabia over ISIS and Iran at a time of strained relations between the countries, making the bill's timing that much more sensitive.

The stepped-up lobbying against the legislation comes as it is coming up against fresh roadblocks on Capitol Hill, with party leaders learning that a GOP senator is objecting to taking up the bill, according to a source familiar with the legislation. The senator's identity has not yet been revealed publicly.

Proponents of the measure, for their part, are beginning to intensify their pressure campaign.

"If Saudi Arabia participated in terrorism, of course they should be able to be sued," Schumer said Monday. "This bill would allow a suit to go forward and victims of terrorism to go to court to determine if the Saudi government participated in terrorist acts. If the Saudis did, they should pay a price."

Speaking to reporters Monday, White House spokesman Josh Earnest fired back, warning that it would jeopardize international sovereignty and put the U.S. at "significant risk" if other countries adopted a similar law.

"It's difficult to imagine a scenario where the President would sign it," Earnest said.

The bill, which Schumer and Senate Majority Whip John Cornyn of Texas are pushing, would prevent Saudi Arabia and other countries alleged to have terrorist ties from invoking their sovereign immunity in federal court.

Saudi Arabia has long denied any role in the 9/11 attacks, but victims' families have repeatedly sought to bring the matter to court, only to be rebuffed after the country has invoked legal immunity allowed under current law.

"It makes minor adjustments to our laws that would clarify the ability of Americans attacked on U.S. soil to get justice from those who have sponsored that terrorist attack," Cornyn said of the bill, which is entitled the Justice Against Sponsors of Terrorism Act.

A hold in the Senate

But despite the effort, it has hit repeated snags on Capitol Hill. The latest: A GOP senator is privately holding up the bill, an obstacle that would require Senate Majority Leader Mitch McConnell to file time-consuming procedural motions to move past. He would need 60 votes to overcome the objection, but it is uncertain whether he will move on the measure before his colleague's issues are ironed out.

No senator has yet claimed responsibility for privately voicing the objection.

Under the rules of the Senate, any one senator can privately inform his or her party leadership of plans to place a "hold" to block legislation. His or her name would not become public unless a senator took to the Senate floor and tried to advance the bill, forcing the opponent to object. That scenario has not yet occurred as negotiations to find a way forward continue behind the scenes.

Cornyn said that while several senators originally had placed holds on the measure, they had all relented save one.

"I think part of the concern is that somehow this is a thumb in the eye to Saudi Arabia, a valuable ally," he said. He then defended the bill, saying, "It's not open-ended and it's not targeted at Saudi Arabia."

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How House and Senate Republican leadership deals with the matter is unclear.

While the bill was easily approved by the Senate Judiciary Committee in January with the support of a cross-section of senators including presidential candidate Ted Cruz, a Texas Republican, and liberals like Minnesota's Al Franken, McConnell has not yet taken a position on the issue, according to a spokesman.

The House is awaiting Senate action before proceeding on a companion bill introduced by Rep. Peter King, a Republican of New York. House Speaker Paul Ryan did not discuss the legislation during his recent trip to Saudi Arabia, according to an aide familiar with the matter.

On Tuesday, Ryan told reporters at a news conference that the bill should be reviewed carefully to see of unintended consequences.

While it may be risky politically for any senator to object to a bill supported by 9/11 victims' families, some are warning that it could have broader ramifications if it were to become law.

Paul Callan, a CNN legal analyst, warned that the bill could impact the U.S. if another country were to retaliate against Americans drone attacks, for instance.

"Which is why for almost 200 years, international law has recognized this concept of sovereign immunity that countries shouldn't really allow individual courts to sue other countries," Callan said on "Legal View" with Ashleigh Banfield. "It shall be worked down as a matter of foreign relations."

Campaign rhetoric intensifies

While Saudi Arabia has never been implicated in the 9/11 attacks, 15 of the 19 hijackers were of Saudi descent. Moreover, there has long been suspicion about ties between the royal family of Saudi Arabia to al Qaeda, speculation that has only intensified as 28 pages of a congressional investigation into the 9/11 attacks remains classified.

As pressure grows on Congress to let 9/11 victims' families pursue their claims against Saudi Arabia in federal court, Saudi officials are quickly pushing back.

In a stark warning to members of Congress, Saudi Foreign Minister Adel al-Jubeir warned lawmakers last month in Washington that his kingdom would sell $750 billion in U.S. assets, including treasury securities, if the measure became law, sources familiar with the matter told CNN. The development was first reported in The New York Times.

Cornyn, however, dismissed the threat.

"It's seems overly defensive to me and I doubt they can do it," he said in response. "I don't think we should let foreign countries dictate the domestic policy of the United States so, no, it doesn't bother me at all."

Presidential candidates were also unmoved. Ahead of the New York primary, former Secretary of State Hillary Clinton and Vermont Sen. Bernie Sanders quickly sought to align themselves with the Cornyn-Schumer bill.

After Clinton said in a Sunday appearance on ABC that she had to study the bill and would not take a position, a spokesman later said she backs the bill.

Sanders, in a statement Sunday night, announced that he supports the bill and called on the Obama administration to declassify the 28 pages of the 9/11 report that could implicate Saudi Arabia.

Other presidential candidates jumped into the fray, including GOP front-runner Donald Trump.

Appearing on the Joe Piscopo Show, a New York radio program, Trump evinced no concern about Saudi Arabia's threat to sell off U.S. assets.

"Let 'em sell 'em," Trump said. "No big deal."

Trump added: "Hey, look, we protect Saudi Arabia. We protect them for peanuts. If we weren't protecting them, they wouldn't be there for a week."