"Mr. Vice President, I asked you earlier if anyone on your staff wrote this op-ed".
He restated that he thinks the essay writer should do the "honourable thing and resign".
The Times op-ed published this week, attributed to a "senior administration official" at the request of the author, described a group of individuals in the White House seeking to thwart the president's "worst inclinations".
Woodward told CBS Morning News on Sunday that "you look at the operation of this White House and you have to say, 'Let's hope to God we don't have a crisis'".More news: Woman on antibiotics develops a 'black hairy tongue' in freakish medical case
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"I would agree to take it in a heartbeat and would submit to any review the administration wanted to do", he told Wallace.
Both Pence and Kellyanne Conway, a counselor to Trump, pushed back during separate television appearances Sunday on the portrayals of Trump as anything but a thoughtful leader.
They also come as former president Barack Obama has stepped forward to harshly criticize Trump and Republican politics, comparing Trump in a speech Friday to demagogues around the world who exploit "a politics of fear and resentment and retrenchment". But he declined to say whether he believes all top officials should be made to do the same, saying it was a decision for Trump to make.
"Does the so-called "Senior Administration Official" really exist, or is it just the Failing New York Times with another phony source?" He cited national security concerns for what would amount to an extraordinary criminal probe should Attorney General Jeff Sessions decide to pursue one.
"People who work for him are anxious - that he will sign things or give orders that threaten the national security or financial security of the world", Woodward said on the "Sunday Morning" broadcast.
Pence also said that he hasn't been asked for an interview by special counsel Robert Mueller as part of the investigation into Russian interference in the 2016 presidential election and added that he was willing to sit down with Mueller if asked.