The American Healthcare Act is intended to replace parts of President Barack Obama's signature "Obamacare" law.
A vote on Thursday was delayed because of opposition from some Republicans.
Mr Trump reportedly warned fellow Republicans that they had a choice between voting for his bill on Friday or being stuck with Obamacare for good.
The president made the warning during a closed-door meeting at the White House, US media reported.
Republican and House Speaker Paul Ryan said: "For seven-and-a-half years we have been promising the American people that we will repeal and replace this broken law because it's collapsing and it's failing families, and tomorrow we're proceeding."
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Meanwhile, Chris Collins, New York's Republican representative, said: "The president has said he wants a vote tomorrow, up or down. If for any reason it is down, we are just going to move forward with additional parts of his agenda."
Repealing and replacing Obamacare was a major plank of Mr Trump's election campaign, but his replacement has stalled amid Republican infighting. The party is unable find a compromise: the current reforms go too far for some and not far enough for others.
The postponement of Thursday's vote was a setback for the president, who had insisted he would win the numbers to pass it through the lower chamber of Congress on that day.
He needs a minimum of 215 Republicans to vote for the bill. If 22 Republicans join the Democrats in voting against the bill it will fail, and an Associated Press tally late on Thursday suggested that at least 28 Republicans opposed it.
The administration's hope is that Mr Trump's ultimatum will force Republicans opposed to the bill to vote Yes if the alternative is the preservation of Mr Obama's healthcare legislation.
Earlier on Thursday, House Minority Leader Nancy Pelosi said Mr Trump had made a "rookie's error for bringing this up on a day when clearly you're not ready".
The bill needs 215 votes to pass but ran into opposition mainly from conservative Republicans who believed it did not roll back enough of Mr Obama's Affordable Care Act.
Obamacare helped 20 million previously uninsured Americans get health insurance but has been plagued by increases in insurance premiums, which were also a problem before the health law.
Mr Trump promised a new law that would cover more people and at a lower cost.
The Republican bill keeps some of the popular elements of Obamacare but limits future federal funding for Medicaid, which covers low-income people.
A new estimate by the Congressional Budget Office released on Thursday evening said recent changes to the bill would make it costlier than previously thought.
The number of uninsured Americans would rise to 24 million by 2026 under the new law, the budget analysis said.
Groups representing doctors, hospitals and the elderly have said they are opposed to the Republican bill.
key elements of the new bill:
- Cuts the Medicaid programme for low earners
- Provides tax credits to help people pay medical bills, but reduced compared to Obamacare
- Ends penalties on those who do not buy health coverage
- Allows insurers to raise premiums for older people
- Blocks federal payments to women's healthcare provider Planned Parenthood for a year