Cobb, who was hired to oversee the White House's legal and media response to the investigation into Russian meddling, was heard talking openly about the Russia investigation with John Dowd, a Washington lawyer with experience in high-profile political cases.
"The White House counsel's office is being very conservative with this stuff," Cobb reportedly told Dowd at BLT Steak in Washington last week. "Our view is we're not hiding anything."
He added White House counsel Don McGahn had "a couple of documents locked in a safe" that he suggested he wanted access to.
Cobb also expressed concern with a White House lawyer he didn't name in the conversation.
"I've got some reservations about one of them," Cobb reportedly said. "I think he's like a McGahn spy."
Cobb also reportedly brought up the June 2016 Trump Tower meeting -- when Donald Trump Jr., Jared Kushner and former campaign chairman Paul Manafort met with Russian operatives after the younger Trump was informed the Russian government was trying to help his father's campaign -- as well as the White House's response to it, saying "there was no perception that there was an exchange."
And he mentioned an unnamed colleague on whom he blamed "some of these earlier leaks," adding the colleague "tried to push Jared out," meaning Kushner, President Donald Trump's son-in-law and senior adviser.
The Times reported that after it contacted the White House about the conversation, sources said McGhan privately "erupted" at Cobb about the situation.
White House chief of staff John Kelly also reprimanded Cobb over the incident, sources told the Times.
New York Times reporter Kenneth Vogel, who overheard the conversation, tweeted a photo of Cobb and Dowd at the next table from him at the restaurant.
CNN has reached out to the White House for comment.
GOP Rep. Chris Collins told CNN's Alisyn Camerota on "New Day" that the report is a reminder public officials shouldn't discuss official business at a restaurant.
"If you're sitting, talking with someone at dinner, you need to be careful," he said. "I would not read anything into this other than it's a word of caution for anyone, but certainly anyone working for the President."