Contradicting President Obama, the State Department acknowledged Thursday that the payment of $400 million to Iran in January was contingent on Tehran’s release of four American hostages.

Asked by a reporter if the U.S. wouldn’t have paid the money until the prisoners were released, State Department spokesman John Kirby replied, “That’s correct.”

The link between the payment — meant to settle an Iranian claim dating back to the 1970s over a blocked arms purchase — and Iran’s decision to release the four Americans, including Washington Post bureau chief Jason Rezaian, has become increasingly controversial in recent days. Mr. Obama and his top aides found themselves under fire for what critics says was a break with the long-standing U.S. practice of not negotiating with terrorist or terrorist-sponsoring regimes for hostages.

While Mr. Kirby didn’t use the word “ransom,” his account differed from Mr. Obama’s assertion Aug. 4 that there was no link between the payment and the release of the hostages.

“We do not pay ransom,” Mr. Obama insisted, saying the two negotiations went along completely separate tracks. “We didn’t here, and we won’t in the future.”

Administration officials also said the $400 million payment — although not the details of how it was carried out — was disclosed in January at the same time as the hostages were being released.

House Speaker Paul D. Ryan, Wisconsin Republican, said Mr. Obama “owes the American people a full accounting of his actions and the dangerous precedent he has set.”

“Today, the State Department admitted what we’ve long suspected — that the president and his administration have been misleading us since January about whether he ransomed the freedom of the Americans unjustly imprisoned in Iran. The president owes the American people a full accounting of his actions and the dangerous precedent he has set,” Mr. Ryan said in a statement.

Republican nominee Donald Trump has repeatedly criticized the payment to Tehran, and Sen. Tom Cotton, Arkansas Republican and a prominent critic of the administration’s dealings with Iran, said the report from the State Department confirms “what many of us have suspected all along: The $400 million payment to Iran was a ransom payment for four Americans unlawfully held by the ayatollahs.”

“And we now know the extraordinary lengths to which the Obama administration went to ensure this payment happened, including lifting sanctions on Iranian Air just one day before the transfer,” Mr. Cotton said in a statement.

He said terrorist groups now will be motivated by the knowledge that “the United States will pay cold, hard cash for hostages.”

Mr. Cotton suggested that the Senate will hold hearings on the matter when lawmakers return to Capitol Hill next month. “The administration will no longer be able to hide from giving Americans straight answers,” he said.

The $400 million payment was delivered to Tehran in bundles of European currency stacked on wood pallets in a cargo plane. The administration has said it was the first installment of a $1.7 billion settlement of a decades-old complaint from Iran over the collapse of a deal struck by the Carter administration with the Shah of Iran shortly before he was ousted in the 1979 revolution.

Mr. Kirby said the U.S. was “able to conclude multiple strands of diplomacy within a 24-hour period” in January, including implementation of the nuclear deal, “the prisoner talks, and a settlement of an outstanding Hague tribunal claim, which saved American taxpayers potentially billions of dollars.”

He said the administration sought to use all its bargaining power at that moment to gain release of the hostages.

“With concerns that Iran may renege on the prisoner release given unnecessary delays regarding persons in Iran who could not be located, as well as, to be quite honest, mutual mistrust between Iran and the United States, we of course sought to retain maximum leverage until after American citizens were released,” Mr. Kirby said.

Republican lawmakers weren’t buying that reasoning, yet again.

“If it quacks like a duck, it’s a duck. If a cash payment is contingent on a hostage release, it’s a ransom,” said Sen. Ben Sasse, Nebraska Republican. “The truth matters, and the president owes the American people an explanation.”

House Financial Services Committee Chairman Jeb Hensarling, Texas Republican, noted that Congress already has requested documents from the Justice Department, the Treasury Department and the Federal Reserve related to the cash payment.

“The State Department’s admission that this money was indeed a ransom payment intensifies the need for Congress to fully investigate this outrageous action by the administration and the urgency for the administration to turn over these records immediately,” Mr. Hensarling said.

The Republican National Committee said the State Department’s explanation Thursday “was the dictionary definition of a ransom payment and a complete contradiction of what they were saying just two weeks ago.”

“It’s time for the Obama White House to drop the charade and admit it paid a ransom to the world’s leading state sponsor of terrorism,” said RNC spokesman Michael Short. “Hillary Clinton’s support for this dangerous blunder, which has put a price on the head of every American citizen, shows once again she does not have the judgment to be president.”

Matt Brooks, executive director of the Republican Jewish Coalition, called on Mrs. Clinton and other Democrats to repudiate the ransom payment and reverse their support for the nuclear deal with Tehran.

“The Obama administration’s ransom payment to the largest state sponsor of terrorism sets a dangerous precedent, one that will only put more Americans at risk,” he said. “This $400 million ransom, along with the $100 billion Iran has already collected from the U.S., will further Iran’s funding of terrorist organizations who launch regular attacks in Israel, as well as their funding of militia fighters directly working to undermine our coalition forces in Iraq and Syria.”

At a press conference two weeks ago, Mr. Obama responded with borderline disgust to the accusations of ransom-paying.

Families “know we have a policy that we don’t pay ransom. And the notion that we would somehow start now, in this high-profile way, and announce it to the world, even as we’re looking in the faces of other hostage families whose loved ones are being held hostage, and saying to them, ‘We don’t pay ransom,’ defies logic,” Mr. Obama said at the time.

The State Department also reported Thursday that it was unable to find the person who requested the editing of a videotaped press briefing that deleted a passage about the Iranian nuclear deal, after an investigation that included interviews of more than 30 current and former department employees.

Mr. Kirby said the probe concluded that the video of former State Department spokeswoman Jen Psaki was “deliberately edited” within 18 minutes of the briefing’s conclusion. But other than that, he said, the department has no answers about who requested the edit or why.

“It’s not impossible or inconceivable that there was an attempt to conceal information — in other words, nefarious intent here. We’re not ruling that out,” Mr. Kirby said. But he added that the edit could have been the result of a “technical problem.”

The portion of the video of a State Department press briefing in December 2013 addressing secret talks between the U.S. and Iran was deliberately deleted before it was posted online. Mr. Kirby said previously that an unknown U.S. official requested over the phone that a technician delete part of the video. But the technician can’t remember who made the request.

After it was revealed that secret talks between the U.S. and Iran had taken place, Ms. Psaki acknowledged that the administration lied to hide those negotiations from the public.

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