Senator Susan Collins, a moderate Republican from Maine, said her colleagues were “starting from scratch.”

“We’re going to draft our own bill,” Ms. Collins said on “This Week.” “And I’m convinced that we’re going to take the time to do it right.”

She and three other Senate Republicans introduced in January their own alternative to the Affordable Care Act that would preserve certain protections, including for those for people with pre-existing conditions. The House bill would allow states to seek waivers for many such provisions, and although it would set aside billions for “high-risk pools,” critics say care could still be unaffordable for those with pre-existing conditions.

Senator Roy Blunt, a Missouri Republican and the chairman of the Senate subcommittee on health and human services, said on NBC’s “Meet the Press” that the Senate would work on its own version of the bill and examine a budget office analysis of it before moving toward a vote.

“This is the way legislation used to be passed,” he said.

Amid the volleys between the House and the Senate, White House officials said on Sunday that the legislative process was moving along as it should. Reince Priebus, the White House chief of staff, who pushed hard to get a revised version of the health care bill through the House, said he had already spoken to six Republicans who are part of a group tasked with writing the Senate’s version of the bill.