House Speaker Paul D. Ryan said Sunday that he feels “very good” about his Obamacare replacement’s prospects for passage this week but that negotiators are reaching for ways to help older Americans whose costs would soar under the plan as written.

Mr. Ryan, Wisconsin Republican, said President Trump is a terrific “closer” who persuaded a dozen conservatives to back the plan. The president will use his negating chops to reel in more Republican holdouts, he said.

Yet finding a way to help Americans ages 50 to 64 remains a key sticking point for Republicans coming to grips with the potential fallout of allowing insurers to charge five times what they charge younger adults.

“That is among the things we’re looking at doing, yes,” Mr. Ryan told “Fox News Sunday.”

Centrist Republicans, whose votes will be needed to pass the plan, have been demanding more tax credit assistance for older constituents after the Congressional Budget Office said their costs could spike by thousands of dollars.

The CBO also estimated that the plan would save more than $300 billion but result in an eye-popping 24 million fewer people being insured a decade from now.

Still, any decision to ramp up refundable tax credits in the plan could scare off conservatives, who say the existing plan amounts to a new entitlement and doesn’t do enough to drive down premiums.

Leaders of the House Freedom Caucus say they have the votes to block the plan unless major changes are made.

Sen. Rand Paul, Kentucky Republican, continued to dub the legislation “Obamacare lite,” foreshadowing an uphill fight in the Senate.

Mr. Paul has objected to the tax credits and a portion of the bill that requires insurers to tack on a 30 percent surcharge for customers who go without coverage for 63 days or more. He said it’s eerily reminiscent of Obamacare’s mandate to hold insurance.

“We never ran on making the entitlement subsidies permanent. We never ran on an individual mandate or keeping the taxes or keeping the Medicaid expansion,” Mr. Paul told ABC’s “This Week.”

“We didn’t run on that, and so they’re really flat-out false in telling us, ‘Oh, you guys ran on this plan.’ None of us ran on this plan. We ran on repealing Obamacare because it doesn’t work,” Mr. Paul said.

The speaker predicted victory in the House, however, saying Mr. Trump is committed to scrapping Obamacare once and for all.

He said the president and Republican lawmakers are fine-tuning the bill and securing votes for the plan as it heads to the Rules Committee to pin down final changes.

“There’s one word to describe what’s going on: legislating,” Mr. Ryan said.

Mr. Ryan said a floor vote most likely would be held Thursday, the seventh anniversary of the Affordable Care Act’s passage. The administration agreed Friday to amendments that would make changes to Medicaid, the insurance program for the poor.

The changes would allow states to impose work requirements on able-bodied people in the program and let governors accept a block grant of federal funding for the program instead of a per-capita allotment.

Because of those changes, a dozen conservative House lawmakers from the Republican Study Committee gave their commitments Friday to Mr. Trump to vote for the bill.

House Majority Whip Steve Scalise, Louisiana Republican, said the changes strengthened the count, but he stopped short of saying leaders had gathered the 216 votes they need.

Mr. Ryan can’t afford to lose more than 21 Republican votes because no Democrats are expected to support the bill, yet he is optimistic.

“I’m very impressed with how the president is helping us close this bill,” Mr. Ryan said.

By contrast, House Minority Leader Nancy Pelosi, California Democrat, said Mr. Trump “either doesn’t know or doesn’t care” what his health care bill would do to his most ardent supporters. She cast the plan as a massive transfer of wealth from red-state voters who will lose coverage to wealthy blue-staters who would receive tax cuts.

“When he says ‘death spiral’ and this and that, he really doesn’t know what he is talking about. It is most unfortunate. And so this product is ‘Trumpcare,’” she told CBS’ Face the Nation.

Vice President Mike Pence on Saturday said the White House is “working around the clock” with Republican lawmakers who have expressed doubts about the plan.

“Every day, more and more members of Congress are getting on board,” Mr. Pence said during a visit to a paper company in Jacksonville, Florida. “We’re counting on Florida. We need every Republican in Florida to support this bill and support the president’s plan to repeal and replace Obamacare.”

Two Republican lawmakers from Florida, Reps. Ron DeSantis and Bill Posey, are Freedom Caucus members.

Senate Republicans are using fast-track budget rules to avoid a Democratic filibuster, yet some Senate Republicans say the House legislation will need major changes to gain their votes.

Like Mr. Paul, Sen. Ted Cruz of Texas said he will not vote for the plan unless it strikes down Obamacare’s insurance regulations.

Mr. Cruz said the House Republicans’ “three-bucket” plan puts too much faith in the Trump administration’s ability to use regulatory authority and future legislation, which can be filibustered, to erase Obamacare entirely.

“Bucket three takes eight Democrats,” he told CBS’ “Face the Nation.” “Right now, Senate Democrats are opposing everything. You can’t get eight Democrats to agree on saying, ‘Good morning.’”

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