WASHINGTON — The Senate Intelligence Committee appears to be moving toward swift passage of a bill that would “change but preserve” the once-secret National Security Agency program that is keeping logs of every American’s phone calls, Senator Dianne Feinstein, the California Democrat who leads the panel, said Thursday.

Ms. Feinstein, speaking at a rare public hearing of the committee, said she and the top Republican on the panel, Senator Saxby Chambliss of Georgia, are drafting a bill that would be marked up — meaning that lawmakers could propose amendments to it before voting it out of committee — as early as next week.

After the existence of the program became public by leaks from the former N.S.A. contractor Edward J. Snowden, critics called for it to be dismantled. Ms. Feinstein said her bill would be aimed at increasing public confidence in the program, which she said she believed was lawful.

The measure would require public reports of how often the N.S.A. had used the calling log database, she said. It would also reduce the number of years — currently five — that the domestic calling log data is kept before it is deleted. It would also require the N.S.A. to send lists of the phone numbers it searches, and its rationale for doing so, to the Foreign Intelligence Surveillance Court for review.