The department said Sessions' staff relied on the guidance of the FBI investigator handling the background check, who advised that meetings with foreign dignitaries "connected with Senate activities" did not have to be reported on the form.
At a hearing Tuesday before the Senate Armed Services Committee, Coats said he couldn't confirm or deny the report.
In a statement, Justice Department spokesman Ian Prior said Sessions met with hundreds, if not thousands, of foreign dignitaries while in the Senate.
Warner said one subpoena had been served and that he expected the second one to be served Tuesday evening.
Last week, CNN published a report alleging that intercepted phone calls showed Russian officials bragged about influencing Flynn and by extension Trump's administration.
Also Wednesday, the FBI told a House committee that it would not be complying with a Wednesday deadline to turn over memos written by former FBI Director James Comey detailing his discussions with President Donald Trump.
Flynn, a top adviser to candidate Trump before becoming national security adviser, was paid more than $33,000 in 2015 to speak at a Moscow gala where he was seated next to Russian President Vladimir Putin.
President Trump has meddled in the Russian Federation investigation and tried to get the investigation dropped against Mike Flynn, but House Intel Committee Ranking Member Rep. Adam Schiff (D-CA) is warning that Flynn may be found in contempt of Congress if he ignores House subpoenas.
Both of the committees want Flynn to give information about Trump's links with Russian Federation and any knowledge about any interference by Russian Federation in the U.S. election.
The two global leaders, vastly different in temperament and views of the world, talked seriously and extensively in a 30-minute private meeting about terrorism, the radicalization of young people, immigration and climate change, officials said. The report Wednesday says the Russians zeroed in on Trump campaign chairman Paul Manafort and Michael Flynn, a former head of US military intelligence who was a key Trump campaign adviser.
A day earlier, former CIA Director John Brennan told a House committee that he had seen intelligence that "revealed contacts and interactions" between Russian officials and Americans "involved" in the Trump campaign - a cause for concern in case the Russians could get the Americans to cooperate.More news: Trump travel ban showdown headed for Supreme Court
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"We will be following up with subpoenas and those subpoenas will be created to maximize our chances of getting the information we need", Rep. Adam Schiff of California, the highest-ranking Democrat on the House Intelligence Committee, said Wednesday. He did not elaborate on what materials the committee was seeking.
The committee is also looking for ways to obtain documentation of Trump's conversations with top intelligence and law enforcement officials involved in a separate federal Russian Federation probe. They also said an "escalating public frenzy" against Flynn and the appointment of a special counsel had created a legally perilous environment for Flynn to provide the information.
"I can only say I have fully complied with their specific request", Stone told The Associated Press in an email Tuesday.
Burr and Warner announced they would subpoena documents from the two Flynn businesses they were aware of - Flynn Intel LLC and Flynn Intel Inc., both based in Alexandria, Virginia.
He will invoke the Fifth Amendment and his right against self-incrimination.
Any immunity agreement, he said, would only take place after speaking with former FBI Director Robert Mueller, the special counsel leading the executive branch's investigation into potential collusion between Russian Federation and Trump's associates.
However, he said there was a sufficient basis of information and intelligence that warranted further investigation by the FBI to determine "whether U.S. persons were actively colluding".
Read the full New York Times report here.
The early months of Trump's presidency have been clouded by Federal Bureau of Investigation and congressional probes into Russian meddling and possible collusion by Trump's campaign.
He said he raised published media reports of Russian attempts to meddle in the election with the Russian official, who denied any involvement by Moscow.