Top aides to President Trump on Monday said they expect the House of Representatives will vote this week to overhaul the US health-care system, but it was unclear when a vote would be scheduled, and moderate Republican lawmakers remained skeptical.

The White House is eager to move forward on legislation to repeal and replace the Affordable Care Act, commonly called Obamacare, to make good on a key campaign promise. Republicans tried but failed to pass a replacement bill in March in an embarrassing setback for the Trump administration.

White House Chief of Staff Reince Priebus and White House economic adviser Gary Cohn on Monday predicted action to unwind former President Barack Obama's health-care law would soon succeed.

"I think it will happen this week," Mr. Priebus said on CBS "This Morning" television program. In a separate interview, Mr. Cohn told CBS the White House was "convinced we've got the votes" in the House.

Republican lawmakers have struggled to unite around legislation, with moderates and conservatives within the caucus divided over key provisions.

"We are having those member-to-members conversations right now," Cathy McMorris Rodgers, chair of the House Republican conference, said on Fox News.

Ms. McMorris Rodgers said Republican members needed time to understand new tweaks to the bill. She declined to elaborate on how close leaders were to having enough support to pass the legislation, saying: "We're very close."

As of Monday morning, no vote had been scheduled, nor have administration officials and backers of the health-care proposal released legislative language.

"We will schedule a vote when we have the needed votes,” said AshLee Strong, a spokeswoman for House of Representatives Speaker Paul Ryan.

Even if a plan could pass the Republican-controlled House, it would face a tough fight in the Senate, where Republicans have a narrower majority and where some party senators have expressed misgivings about the Republican bill.

At issue is an amendment that will provide more flexibility to states to run health-care programs. States could seek waivers from some of Obamacare's most popular provisions, including one mandating that insurers charge those with pre-existing conditions the same as healthy consumers.

The conservative House Freedom Caucus, which brought down previous efforts to pass a bill, has endorsed the measure. But several moderate Republicans were either undecided or opposed the bill for fear that it would not protect those with pre-existing conditions and cause millions to lose health insurance.

Republican Representative Charlie Dent, a moderate from Pennsylvania, said he still had problems with the latest version of the plan and suspected there were not enough votes to pass it now.

"Too many Americans are going to be without coverage," Mr. Dent told MSNBC, adding that the current plan could make things even worse for vulnerable Americans.

"Nothing has changed as far as I know," said one moderate GOP aide. "Our sense is we're not seeing a lot of moderates who are looking at this."

Support from moderates will likely be needed to get the 216 votes in the House of Representatives Republicans need for passage.

Mr. Trump told CBS News in an interview over the weekend that pre-existing conditions would be covered in the bill, but said health care decisions should rest with states. On Sunday, he tweeted that a "new healthcare plan is on its way. Will have much lower premiums & deductibles while at the same time taking care of pre-existing conditions!"

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House Freedom Caucus Chairman Jim Jordan, in several television interviews on Monday, said he expected there would be enough House Republican votes to pass the bill this week.

"This bill doesn’t get all the way there but it’s a good step and is ... the best we can get out of the House right now," Mr. Jordan told CNN.