“There are not the votes in the House to pass a clean CR,” the Ohio Republican said on ABC’s “This Week,” in his first extended interview since the government shutdown began Tuesday.

But Democrats weren’t buying it.

“That flies in the face of all the math and public statements of GOP members,” Obama senior adviser Dan Pfeiffer said on Twitter. “Either he is all wrong or all his members are lying.”

Boehner’s comments contradict whip counts done by news organizations and by Democrats, which have found at least 21 Republicans willing to join with the vast majority of Democrats to pass a clean continuing spending resolution.

( WATCH: John Boehner: ‘It is time for us to stand and fight’)

Democrats’ response? We dare you to just hold the vote.

“Why doesn’t he put it on the floor and give it a chance?” Treasury Secretary Jack Lew asked on CBS’s “Face the Nation.” I worked for Speaker [Tip] O’ Neill who believed deeply the one thing he believed the American people won’t tolerate is obstructionism. He put things on the floor, and sometimes he won and sometimes he lost. But that’s the right thing to do.”

“If he’s right, why not prove it?” White House press secretary Jay Carney echoed on Twitter.

Sen. Chuck Schumer (D-N.Y.) called it a “friendly challenge” for Boehner.

“Put it on the floor Monday or Tuesday,” he said on “This Week,” just after the speaker appeared on the show. “I would bet there are the votes to pass it. We have just about every Democrat, 21 Republicans have publicly said they would. There are many more Republicans who have said that they privately would.”

Boehner spokesman Michael Steel didn’t respond to a question about whether the speaker would take Democrats up on their continuing resolution challenge, instead turning attention to the debt ceiling. “As part of the Senate Democratic leadership, Sen. Schumer ought to concentrate on passing a clean debt limit increase in the Senate,” Steel said. “That’s President Obama’s position, so why hasn’t the Senate voted on it?”

Obama has insisted that he won’t negotiate to end the government shutdown or to raise the debt ceiling, arguing that Republicans are trying to “extract a ransom” just for doing their jobs. To Boehner, Obama is the intransigent one. “It’s my way or the highway,” the speaker said. “Complete surrender and then we’ll talk to you.

“It is time for us to stand and fight,” Boehner declared.

That fight encompasses not just the give-and-take to fund the government, but also to raise the debt ceiling, something the administration says must be done by mid-October.

“I’m telling you that on the 17th, we run out of our ability to borrow, and Congress is playing with fire,” Lew said on CNN’s “State of the Union,” one of four Sunday shows on which he appeared. “If they don’t extend the debt limit, we have a very, very short window of time before those scenarios start to be played out.”

Countering on NBC’s “Meet the Press,” Sen. Rand Paul (R-Ky.) charged it was “irresponsible” for Obama and his aides to discuss the debt ceiling in such dire terms.

But Boehner said that on the debt ceiling, too, an increase will come only after negotiations with Democrats.

“We’re not going to pass a clean debt limit increase,” he said. “I told the president, there’s no way we’re going to pass one. The votes are not in the House to pass a clean debt limit. And the president is risking default by not having a conversation with us.”

Though some, including Obama, have cast what happened as Boehner being forced into a position by the more conservative members of his conference, the speaker said he was in charge. “I, working with my members, decided to do this in a unified way,” he said, with demands to defund, delay or otherwise alter Obamacare.

Boehner said he had expected that the health care fight would come during the next vote to raise the debt ceiling. But, he said, “You know, working with my members, they decided, let’s do it now…And this fight was going to come, one way or another. We’re in the fight. We don’t want to shut the government down. We’ve passed bills to pay the troops. We passed bills to make sure the federal employees know that they’re going to be paid throughout this.”

But Sen. Ted Cruz (R-Texas), the freshman who’s jokingly been called “speaker of the House” for helping push the chamber’s tea-party Republicans toward attaching Obamacare demands to the continuing resolution, suggested his sights have moved on to the debt ceiling deal, or perhaps to a broader agreement that reopens the government, raises the debt ceiling and exacts concessions from Democrats on Obamacare.

“We should look for ways to mitigate the harms from Obamacare,” he said on “State of the Union.” “The debt ceiling historically has been among the best leverage that Congress has to rein in the executive.”

Tarini Parti contributed to this report.