The daily was one of several media outlets that reported that Trump himself was being investigated on allegations of obstructing justice.
The committee sent a bipartisan letter this month to White House counsel Don McGahn seeking an answer by this Friday.
Some of his ire is aimed at Rod Rosenstein, the United States deputy attorney general and investigative special counsel Robert Mueller, both of whom the president believes are biased against him, associates say.
The president wrote on Twitter Friday: "I am being investigated for firing the FBI Director by the man who told me to fire the FBI Director!" Schiff said if the panel can't get an answer then he believes a subpoena will be needed.
Sekulow said Trump "is not and has not been under investigation".
"Witch hunt" is how Trump characterizes the probe into Russia's election interference and possible ties to his campaign associates.
"That signifies this president knew all too well that it was inappropriate", Schiff said.
Rubio is a member of the Senate Intelligence Committee, which is doing its own probe into Russian election interference. King said the "collusion, the cooperation aspect of the investigation is not over". A lot of people have said, 'When do you think you'll be done?' Maybe the end of the year. "There's a lot of information yet to go".
"The MAKE AMERICA GREAT AGAIN agenda is doing very well despite the distraction of the Witch Hunt", the Republican leader tweeted while away on his first weekend to Camp David, the official U.S. presidential retreat in the mountains of western Maryland.
And on Sunday an attorney on president's legal team on "Meet The Press" adamantly denied that the president is under any kind of investigation, claiming the message was simply referring to a Washington Post report alleging that Special Counsel Robert Mueller's investigation now includes an obstruction of justice probe of the president.
He spoke on ABC's "This Week".