WASHINGTON — President Trump again succeeded in turning back bipartisan congressional efforts to rebuke his lock-step support for Saudi Arabia after the Senate on Monday failed to override his veto on a series of measures that would have blocked billions of dollars of arms sales to the Persian Gulf region.

The failure is the second time in recent months that the Senate has been unable to muster enough opposition to beat a veto on a rebuke to the administration’s relationship with Saudi Arabia. That underscored that legislators’ simmering anger over both the kingdom’s murder of a dissident columnist and its bloody war in Yemen has its limits.

The measures, introduced by Senator Robert Menendez of New Jersey, the top Democrat on the Foreign Relations Committee, sought to stymie the administration’s effort to circumvent Congress to sell munitions to the gulf nations by declaring an emergency over Iran. Mr. Menendez and other lawmakers had blocked some of those sales, but by declaring an emergency, the administration was able to blow through the blockades.

Mr. Trump, in his veto message, said the legislation “would weaken America’s global competitiveness and damage the important relationships we share with our allies and partners.” He vetoed a separate measure in April that would have forced an end to American military involvement in the Yemen war.

The defeat on Monday left lawmakers who are intent on penalizing the administration’s relationship with the kingdom pledging to redouble their efforts. Democratic senators and a handful of Republicans — unwilling to let the issue go but seemingly unable to expand their ranks of supporters — are pressing on, bitterly dividing the Foreign Relations Committee.

“It’s difficult to accept Saudi Arabia as a trusted ally of the United States right now,” said Senator Christopher S. Murphy, Democrat of Connecticut. “Instead of holding the Saudis accountable, this administration instead continues to provide them a blank check.”

The committee advanced legislation last week, led by Mr. Menendez and Senator Todd Young, Republican of Indiana, that would impose mandatory sanctions on those found responsible for the murder of Jamal Khashoggi, the Virginia-based Washington Post columnist. The bill would also prohibit American refueling of Saudi coalition aircraft engaged in the civil war in Yemen and ban certain weapons sales to the kingdom. It is unlikely that Senator Mitch McConnell of Kentucky, the majority leader, will bring the legislation up for a vote.

But the passage of the legislation through committee over the chairman’s wishes was remarkable. It was the culmination of a bitter internal fight between Senator Jim Risch of Idaho, the chairman and a Trump ally, and Mr. Menendez, who offered competing bills outlining a starkly different vision of how Congress should address the American relationship with Saudi Arabia.

Mr. Risch’s legislation would have directed the secretary of state to conduct a review of the relationship and deny or revoke visas to some members of the Saudi royal family as punishment for the kingdom’s human rights violations. The legislation would give Riyadh a chance to reset the kingdom’s ties with the United States, Mr. Risch argued, and because it was developed in consultation with the White House and the State Department, it would have a chance at winning the president’s signature.

“We can either send a messaging bill to the president for a veto or we can send legislation that will drive and, more important, form foreign policy,” he said. 

In the end, three Republicans on the panel, Senator Lindsey Graham of South Carolina, Senator Rand Paul of Kentucky and Mr. Young, broke ranks to support Mr. Menendez’s more punitive bill, and Mr. Risch withdrew his legislation from consideration.

“No hard feelings to anyone, it’s not sour grapes,” Mr. Risch said, “but I’m interested in spending time on something we can actually do.”

House Democrats on the Oversight and Reform Committee, however, have made clear that they will continue their investigation into the administration’s support of Riyadh despite the White House’s refusal to cooperate. In a report released on Monday, they focused on Thomas J. Barrack Jr., a top campaign fund-raiser, inaugural chairman and close friend of Mr. Trump, and concluded that he pushed a proposal to circumvent the normal policymaking process and build dozens of nuclear power plants in Saudi Arabia that experts worried could spread nuclear weapons technology in the volatile Middle East.

The New York Times reported this week that federal prosecutors were investigating whether Mr. Barrack violated the law requiring people who try to influence American policy or opinion at the direction of foreign governments or entities to disclose their activities to the Justice Department.

The committee also concluded that IP3, a private company lobbying the White House to transfer United States nuclear technology to Saudi Arabia, has had “unprecedented access to the highest levels” of the administration, including meeting directly with Mr. Trump. It also said the company had lobbied the administration not to require Saudi Arabia to agree to the “gold standard,” a commitment not to use American nuclear technology to make