The White House believed it had an agreement with the House Intelligence Committee to limit questions for Steve Bannon only to events on the presidential campaign, and not during the ousted former chief strategist's time in the Trump administration, an official told CNBC.

According to the White House official, who spoke on the condition of anonymity, staffers for the committee and the White House on Friday discussed the parameters of Bannon's testimony. The White House emerged from that conversation believing it had an agreement to limit the questioning of Bannon just to events during the campaign, and not during the transition period or his time in the White House.

Then, hours into Bannon's closed-door testimony on Tuesday, Bannon's lawyers informed the White House from Capitol Hill that the questions would extend beyond the scope of what the White House understood the agreement to be. At that point, the White House told Bannon not to answer any further.

"We said 'Hey, hey, pump the brakes,'" the official said Wednesday. "We said to Bannon, 'Don't answer those questions because we haven't agreed to that scope under the process.'"

The committee's ranking Democrat, Rep. Adam Schiff, said the White House did not inform him or his fellow party members on the panel about such an agreement.

"That was never conveyed to the minority," Schiff said Tuesday night. "And I think it came as quite a surprise to our entire committee frankly, apart from the majority staffer that was involved in that communication."

The Trump administration official declined to say who initiated the mid-testimony phone call or who took part on behalf of the White House.

At that point, House Intelligence Committee Republicans and Democrats joined forces to issue Bannon a subpoena on the spot to compel his testimony. It is not clear what, if any, questions Bannon answered after that. A Reuters report said the former top Trump aide refused to comply with the subpoena.

Bannon did not comment substantively Tuesday evening as he left Capitol Hill.

The House Intelligence Committee didn't immediately respond to a request for comment from CNBC.

Despite the attempt to limit questioning of Bannon, the White House official insisted the administration is not asserting executive privilege. "We're not asserting anything," the official said. "They need to discuss it with us. There's a process that's existed for decades."

The official would not say what the White House wants to discuss with the committee or what questions about Bannon's time on the transition or in the White House the administration would seek to block.

Still, press secretary Sarah Huckabee Sanders used the phrase "executive privileges" when asked Tuesday whether the White House was blocking Bannon from testifying fully.

"Look, we've been completely cooperative throughout this entire process," Sanders said. "We're going to continue to be cooperative. But we're also going to maintain some of the executive privileges here at the White House."

In essence, the White House is hoping to reap many of the benefits of executive privilege, without President Donald Trump officially asserting the privilege. Formally asserting executive privilege could be embarrassing for the White House, which has insisted it has nothing to hide in the ongoing Russia investigation and that it is cooperating fully.

Rep. Trey Gowdy, R-S.C., one of the Republicans leading the committee's investigation, reacted with exasperation to Bannon and the White House's claims.

"It is the most tortured analysis of executive privilege I have ever heard of," Gowdy said on Fox News. "Executive privilege now covers things before you become the chief executive — which is just mind-numbing and there is no legal support for it."

Democrats were also frustrated with the 10-hour Bannon meeting.

"This was effectively a gag order by the White House preventing this witness from answering almost any question concerning his time in the transition or the administration and many questions even after he left the administration," said Schiff, D-Calif.

The committee had earlier rebuffed a White House offer Friday to have a White House attorney sit in on the Bannon session to referee the questions surrounding scope of the interview. "The committee's belief was it was not necessary," the official said.

Asked if negotiations over Bannon's testimony are ongoing as of Wednesday morning, the White House official said: "There's no negotiation now, they haven't engaged with us."

But committee members say they still want to hear from Bannon.

"We have additional questions," said Rep. Mike Conaway, R-Texas. "The subpoena remains in effect. And we have additional questions we don't have the answers to yet. We're going to work to get those answers."