"Because then he has to tell what actually took place at the events," Trump told "Fox &Friends" in an interview that aired on Friday.
"My story didn't change. My story was always a straight story. My story was always the truth. But you'll have to determine for yourself whether or not his story changed," he added.
Comey could not immediately be reached for comment via the FBI press office.
On Thursday, Trump tweeted that he did not make, and does not have, recordings of his conversations with the former FBI director. He said he had "no idea" if any such recordings exist.
Trump himself started the speculation about whether he recorded conversations with Comey by suggesting in a tweet last month that there may be recordings.
The possible existence of tapes became more significant after Comey testified that Trump asked him for loyalty and said Trump requested to "drop" a probe into former national security advisor Michael Flynn.
In the Fox interview, Trump also insisted he had not obstructed the FBI's investigation into possible collusion by his campaign with Russia in the 2016 election.
"Look, there has been no obstruction, there has been no collusion," Trump said.
—Reuters contributed to this report.