The meeting came as Mr. Trump was openly feuding with American intelligence agencies and their conclusion that Russia had tried to disrupt the presidential election and turn it in his favor.
The Senate Intelligence Committee notified the White House in March that it planned to question Mr. Kushner about the meeting.
On Friday, citing American officials briefed on intelligence reports, The Washington Post reported that Mr. Kislyak told his superiors in Moscow that Mr. Kushner had proposed a secret channel and had suggested using Russian diplomatic facilities in the United States for the communications. The White House has not denied the Post report, which specified that Russian communication centers at an embassy or consulate in the United States were discussed as hosts for the secure channel.
It is not clear whether Mr. Kushner saw the Russian banker as someone who could be repeatedly used as a go-between or whether the meeting with Mr. Gorkov was designed to establish a direct, secure communications line to Mr. Putin.
The reasons the parties wanted a communications channel, and for how long they sought it, are also unclear. Several people with knowledge of the meeting with Mr. Kislyak, and who defended it, have said it was primarily to discuss how the United States and Russia could cooperate to end the civil war in Syria and on other policy issues. They also said the secure channel, in part, sought to connect Michael T. Flynn, a campaign adviser who became Mr. Trump’s first national security adviser, and military officials in Moscow.
Mr. Flynn attended the meeting at Trump Tower with Mr. Kislyak.
Yet one current and one former American official with knowledge of the continuing congressional and F.B.I. investigations said they were examining whether the channel was meant to remain open, and if there were other items on the meeting’s agenda, including lifting sanctions that the Obama administration had imposed on Russia in response to Moscow’s annexation of Crimea and its aggression in Ukraine.
During the Trump administration’s first week, administration officials said they were considering an executive order to unilaterally lift the sanctions, which bar Americans from providing financing to and could limit borrowing from Mr. Gorkov’s bank, Vnesheconombank. Removing the sanctions would have greatly expanded the bank’s ability to do business in the United States.
In a statement on Monday, Ms. Hicks said that “Mr. Kushner was acting in his capacity as a transition official” in meeting with the Russians. Mr. Kushner has agreed to be interviewed by congressional investigators about the meetings, she said.
In March, Mr. Gorkov said in a statement that his December meeting with Mr. Kushner was part of the bank’s strategy to discuss promising trends and sectors with influential financial institutions in Europe, Asia and the United States. That statement said he met with representatives of “business circles of the U.S., including with the head of Kushner Companies, Jared Kushner.” At the time, Mr. Kushner was still running the company, which is his family’s real estate business.
Vnesheconombank has not responded to questions about which other financial institutions and business leaders Mr. Gorkov met with while in the United States.
Trying to set up secret communications with Mr. Putin in the weeks after the election would not be illegal. Still, it is highly unusual to try to establish channels with a foreign leader that did not rely on the government’s own communications, which are secure and allow for a record of contacts to be created.
But the Trump transition was unique in its unwillingness to use the government’s communications lines and briefing material for its dealings with many foreign governments, partly because of concern that Obama administration officials might be monitoring the calls.
In addition, Mr. Kushner disclosed none of his contacts with Russians or any other foreign officials when he applied for his security clearance in January. He later amended the form to include several meetings, including those with Mr. Kislyak and Mr. Gorkov, but it is unclear whether he told the investigators who conducted his background check about the attempts to set up a back channel. His aides have said his omissions from the clearance form were accidental.
The meeting with Mr. Gorkov is now being scrutinized by the F.B.I. as part of its investigation into alleged Russian attempts to disrupt last year’s presidential campaign, and whether any of Mr. Trump’s advisers assisted in such efforts.
His bank is controlled by members of Mr. Putin’s government, including Prime Minister Dmitri A. Medvedev. It also has long been intertwined with Mr. Putin’s inner circle: It has been used by the Russian government to bail out oligarchs close to Mr. Putin, and has helped fund the Russian president’s pet projects, such as the Winter Olympics in Sochi in 2014.
Vnesheconombank has also been used by Russian intelligence to plant spies in the United States. In March 2016, an agent of Russia’s foreign intelligence service, known as the S.V.R., who was caught posing as an employee of the bank in New York, pleaded guilty to spying against the United States.
The spy, said Preet Bharara, then the United States attorney in Manhattan, was under “the guise of being a legitimate banker, gathered intelligence as an agent of the Russian Federation in New York.”
Mr. Gorkov is a graduate of the academy of the Federal Security Service of Russia, a training ground for Russian spies. Though current and former Americans said it was unlikely that Mr. Gorkov is an active member of Russian intelligence, they said his past ties to the security services in Moscow were a reason he was put in charge of the bank.
But days earlier, responding to questions from The Times about the meetings with Mr. Kislyak and Mr. Gorkov, Ms. Hicks said the meetings were part of an effort by Mr. Kushner to improve relations between the United States and Russia, and to identify areas of possible cooperation.
After the first meeting with Mr. Kislyak, she said at the time, the Russian ambassador asked for a follow-up discussion to “deliver a message.” Mr. Kushner sent Avrahm Berkowitz, a longtime associate and now a White House aide. At that session, Mr. Kislyak told Mr. Berkowitz that he wanted Mr. Kushner to meet Mr. Gorkov, Ms. Hicks said.
Ms. Hicks did not say at the time why Mr. Kislyak had wanted to arrange a meeting between Mr. Kushner and Mr. Gorkov. But she said then that during Mr. Kushner’s meeting with Mr. Gorkov, there was no discussion about the Kushner company’s business or about American sanctions against Russian entities like Vnesheconombank.