In his "Behind the Curtain" column, Woodward, a prolific political author, said Gene Sperling, economic adviser to President Obama, messaged him that he would "regret" questioning the administration's version on how the forced federal spending cuts known as sequestration came about.
The $85 billion in across-the-board federal budget cuts go into effect over the next few months if an agreement is not reached by Friday.
Woodward said in his 2012 best-selling book "The Price of Politics" that the idea for the sequester began in the White House, something the president denied and that the White House later acknowledged.
"They're not happy at all," Woodward said Wednesday on CNN's "The Situation Room." "It was said very clearly, you will regret doing this."
Woodward and Sperling have known each other for decades, beginning when Sperling was an economic adviser in the Clinton administration.
Politico obtained emails exchanged between the men Feb. 22-23 in which Sperling apologized for "raising my voice in our conversation today."
He said the two just did "not see eye to eye here."
"But I do truly believe you should rethink your comment about saying that Potus [president of the United States] asking for revenues is moving the goal post. I know you may not believe this, but as a friend, I think you will regret staking out that claim. The idea of that the sequester was to force both sides to go back to try at a big or grand bargain with a mix of entitlements and revenues..."
Woodward messaged back, "Gene: You do not ever have to apologize to me. You get wound up because you are making your points and you believe them. This is all part of a serious discussion. I for one welcome a little heat; there should be more given the importance."