Trump has hired Kasowitz to handle matters related to various investigations of Russian interference in the USA president election, freeing his White House counsel to handle other government-related matters.
The Republican president stopped short of saying that Mr Comey lied under oath. Comey told members of the Senate Intelligence Committee that he had hoped to prod the Justice Department to appoint a special counsel to oversee the Russian Federation investigation, which occurred last month.
Ms Collins says Comey's motivation "may have been a good one", but says Comey should have given that document to the Senate committee and that Congress should obtain any tapes that Mr Trump might have of his conversations with the ousted Federal Bureau of Investigation chief.
"Well, I didn't say that", Trump answered. He told reporters that he would tell them that "maybe sometime in the very near future".
Earlier in his press conference, Trump said he was "happy" with parts of Comey's testimony, including that the former FBI director said he had assured the president he wasn't personally under investigation regarding the Russian meddling in the election and possible collusion by Trump campaign aides.
He, however, said that whatever Trump did was "a very disturbing thing, very concerning".
"I hardly know the man", Trump said on Friday.
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Comey, who was sacked by Trump in May, did not make any major disclosures about any links between Trump or his associates and alleged Russian meddling.
The committee's chairman, Rep. Mike Conaway Texas, and the ranking Democrat, Rep. Adam Schiff of California, have asked White House Counsel Don McGahn whether any such tapes or memos exist now, or had existed in the past, according to The Associated Press. "Mr. Comey has now admitted that he is one of the leakers".
During the news conference on Friday, ABC News' Jonathan Karl asked Trump to clarify his response to Comey's testimony.
Former Director of Federal Bureau of Investigations (FBI) James Comey said Thursday that he was confused over the circumstances under which he was sacked.
He further denied ever asking Mr Comey for his "loyalty", contradicting Mr Comey's detailed sworn testimony about a private dinner the two men had in the White House.
"I found him to be credible, candid and thorough", Collins said of Comey on MSNBC's "Morning Joe". Trump has not been tweeting anything since June 7.