Donald Trump and the National Rifle Association smoothed over a hiccup in their alliance Sunday, agreeing that the government’s terrorist watch lists are unreliable and should not be used to revoke Second Amendment rights, a position also being taken by liberal-leaning civil liberties groups not usually allied with the NRA or Mr. Trump.

The presumptive Republican presidential nominee and the NRA synchronized their views as the U.S. Senate prepared to vote on Democrat-backed legislation that would do just that. The bill by Sen. Dianne Feinstein of California would ban firearm sales to people on a terrorist watch list or no-fly list. The bill is one of four gun control measures going before the Senate on Monday in response to the terrorist attack last week in Orlando.

All four measures — two Republican and two Democratic — are expected to fall short of the 60 votes needed to advance. But the Orlando bloodshed and fresh calls for more gun control laws from President Obama promised to keep the issue at the forefront in Washington and on the campaign trail.

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“There is not a difference between what Mr. Trump is saying and what the NRA’s position is. That’s a media-created diversion there,” NRA chief lobbyist Chris Cox said on ABC’s “This Week.”

“The FBI should investigate every single person who’s on a terrorist watch list if they try to buy a gun. That’s what they’re doing now,” he said. “If there’s a reason to believe in probable cause that they’re engaged in terrorist activity, they ought to not only be prevented from getting a firearm; they ought to be arrested.”

Mr. Trump raised eyebrows among gun rights advocates when, in response to the Orlando massacre, he pledged to work with the NRA regarding gun sales to people on the government’s famously inaccurate no-fly list.

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Mr. Cox insisted that they held the same position about denying gun rights without due process, which is not provided to people on the watch lists.

In a separate interview on the same program, Mr. Trump said the NRA had the correct approach.

“Some people are on the terrorist watch list who shouldn’t be on it,” he said. “The NRA just has the absolute best interest of our country. They just want to make the right decision. These are great people, great Americans.”

The Senate is taking up the four gun bills after a 15-hour filibuster initiated Wednesday by Sen. Christopher Murphy, Connecticut Democrat.

In addition to the Democrats’ watch-list bill, the Senate will vote on legislation by Senate Majority Whip John Cornyn, Texas Republican, that would delay sales to people on watch lists for up to three days, giving the Justice Department a chance to convince a judge to block the sale.

The bill also would allow authorities to arrest the terror suspect if a judge determines probable cause.

“It would not only stop terrorists from getting guns, but it would take them off the streets, and it would do so in a way that’s consistent with our Constitution,” said Mr. Cornyn, whose bill has the support of the NRA.

Presumptive Democratic nominee Hillary Clinton has made tougher gun laws a cornerstone of her run, and her campaign slammed Mr. Trump last week for not going far enough.

“Donald Trump is trying to soften his tone while likely standing in the way of real progress,” said Clinton campaign spokesman Brian Fallon.

Still, skepticism of the terrorist watch lists is not confined to gun rights proponents.

For years, civil liberties groups have drawn attention to what they say are low standards for inclusion on the secret lists. And they balk at proposals that would deny the ability to challenge a watch list.

“The no-fly list and the terror watch lists can be unreliable, as we’ve seen in the past,” said Jason Pye, spokesman for the libertarian advocacy group FreedomWorks. “Preventing someone from exercising a constitutional right without due processes is frightening. There has to be due process.”

The alliance of special interests, ranging from traditionally conservative gun rights organizations to more socially liberal civil rights groups, “makes for odd bedfellows,” Mr. Pye said.

An American Civil Liberties Union spokesman said the group is still evaluating the two “no fly, no buy” proposals put forth last week by Mr. Cornyn and Ms. Feinstein. But the group has previously battled the Justice Department over its procedures for placing individuals on the no-fly list and notifying them of their inclusion.

In a still-pending lawsuit, the ACLU argued that 13 individuals placed on the list were denied due process because they were never told why they ended up on the list or given an opportunity to be removed.

The government did in 2015 begin informing U.S. citizens of their status on the list, but the ACLU contends that the process is still unconstitutional because the government “withholds evidence and exculpatory information from our clients and refuses to give them a live hearing to establish their credibility or cross-examine witnesses.”

More than 800,000 people were on the FBI’s Terrorist Screening Database, and approximately 64,000 people were on the no-fly list as of September 2014, according to the most recently available data.

As debate has shifted to whether the no-fly list should be used to block firearms purchases, ACLU officials have maintained that the standards for inclusion on the list remain “unconstitutionally vague.” But they haven’t completely shot down the notion of using the list to curtail gun sales in the future if those concerns are addressed.

“There is no constitutional bar to reasonable regulation of guns, and the No Fly List could serve as one tool for it, but only with major reform,” wrote Hina Shamsi, director of the ACLU’s National Security Project, in a December assessment.

Approximately 47,000 people were on the no-fly list in 2013, according to documents obtained by The Intercept. But despite the controversy surrounding the list, it appears to have grown.

Testifying before Congress in late 2014, Terrorist Screening Center Director Christopher M. Piehota said that of 800,000 people on the Terrorist Screening Database, about 8 percent — 64,000 people — were on the no-fly list.

Broader than the no-fly list, the database contains information about people known or “reasonably suspected” of being involved in terrorist activity.

A report from the Government Accountability Office, which Ms. Feinstein released last week, found that known or suspected terrorists initiated background checks to purchase a weapon 2,477 times between February 2004 and December 2015, and they were successfully able to pass the background checks and were approved for firearms purchases in 2,265 instances.

More recently, just 21 of 244 checks conducted of watch-listed individuals through the FBI’s National Instant Criminal Background Check System were denied in 2015.

While the dueling proposals either prevent or delay individuals on watch lists from buying firearms, the gunman who stormed the Orlando gay nightclub was not on a watch list at the time he purchased his guns.

Omar Mateen, who killed 49 people in the deadliest terror attack on U.S. soil since 9/11, was placed on a terror watch list in 2013 while he was under investigation by the FBI.

But shortly after the 10-month FBI investigation was closed, Mateen was removed from the list. He purchased the firearms used in the June 12 mass shooting just days before the attack.

The Senate also is scheduled to vote on a bill by Mr. Murphy and other Democrats to expand background checks and close the so-called gun show loophole.

The final Republican gun bill, sponsored by Sens. Chuck Grassley of Iowa and Ted Cruz of Texas, would add mental health reports to the current background checks and increase funding for the National Instant Criminal Background Check System.