WASHINGTON — Special counsel Robert Mueller has recalled for questioning at least one participant in a controversial meeting with a Kremlin-connected Russian lawyer at Trump Tower in June 2016, and is looking into President Donald Trump’s misleading claim last year that the discussion focused on adoption, rather than an offer to provide damaging information about Hillary Clinton.

Some defense lawyers involved in the case view Mueller’s latest push as a sign the investigators are focusing on possible obstruction of justice by Trump and several of his closest advisers for their statements about the politically sensitive meeting, rather than for collusion with the Russians.

Investigators also are exploring the involvement of the president’s daughter, Ivanka Trump, who did not attend the half-hour sit-down June 9, 2016, but briefly spoke with two of the participants, a Russian lawyer and a Russian-born Washington lobbyist.

Details of the encounter were not previously known.

It occurred at the Trump Tower elevator as the Russian lawyer, Natalia Veselnitskaya, and the lobbyist, Renat Akhmetshin, were leaving the building and consisted of pleasantries, a person familiar with the incident said.

But Mueller’s investigators want to know every contact the two visitors had with Trump’s family members and inner circle.

Mueller long has sought to nail down details of the unusual gathering, at the height of the presidential race, between three of Trump’s top campaign aides — his eldest son, Donald Trump Jr., his son-in-law, Jared Kushner, and his campaign chair, Paul Manafort — and Veselnitskaya and Akhmetshin, plus a Russian language translator, a U.S.-based employee of a Russian real estate group and a British music promoter with Russian business ties who helped bring the group together.

After The New York Times first reported the meeting last July, 13 months after it had occurred, the White House issued a misleading statement while Trump flew back to Washington from the G-20 summit in Hamburg, Germany.

It said that Trump Jr. had said he and the Russian lawyer had “primarily discussed a program about the adoption of Russian children” and was unrelated to the campaign.

Mueller’s team is trying to determine if Trump and others involved in drafting the language aboard Air Force One knew it was inaccurate and whether it was aimed at deceiving federal investigators looking into whether the Trump campaign actively assisted a Russian intelligence operation aimed at interfering in the U.S. campaign.

In August, Trump Jr. released emails that showed he had agreed to the meeting, not to talk about adoptions, but because the music promoter, Rob Goldstone, had assured him that the Russian lawyer had “official documents and information” that would “incriminate” Clinton, “and be very useful to your father.”

Goldstone wrote that the damaging information on Clinton was “part of Russia and its government’s support for Mr. Trump.”

“If it’s what you say, I love it,” Trump Jr. wrote back.

The meeting of top campaign aides and Russians hit the headlines again this week after Steve Bannon, Trump’s former chief strategist, was quoted describing Trump Jr., Kushner and Manafort in scathing terms in a new book.

“Even if you thought that this (meeting) was not treasonous, or unpatriotic, or bad ..., and I happen to think it’s all of that, you should have called the FBI immediately,” Bannon said, according to “Fire and Fury,” released Friday.

Bannon speculated that Trump Jr. brought the Russian group upstairs to meet the candidate, a claim that the participants have denied.

“The chance that Don Jr. did not walk these jumos up to his father’s office on the twenty-sixth floor is zero,” Bannon is quoted as telling the author, Michael Wolff, using Spanish slang meaning drunkards.

The book also says Mark Corallo, then spokesman for the president’s private legal team, quit because he believed the drafting of Trump Jr.’s statement may have obstructed justice.

“The persistent Trump idea that it is not a crime to lie to the media was regarded by the legal team as at best reckless and, in itself, potentially actionable: an explicit attempt to throw sand into the investigation’s gears,” Wolff wrote.

“Later that week, Corallo, seeing no good outcome — and privately confiding that that he believed the meeting on Air Force One represented a likely obstruction of justice — quit.”

Corallo, the Justice Department spokesman from 2002 to 2005, did not respond to requests for comment Friday.

In closed-door testimony before the House intelligence committee last month,Trump Jr. denied that he had communicated directly with his father about the White House statement on adoptions, CNN reported.

The Washington Bureau is not identifying the individual Mueller has asked to return for further questioning as part of an agreement to keep the identity confidential.

david.cloud@latimes.com