In an interview with The New York Times, Trump argued that "Obamacare" has been "catastrophic" and hoped Congress will end it in the coming days.
His replacement, according to Trump, should arrive "very quickly or simultaneously," within a period of less than "weeks."
The president-elect's message comes as some lawmakers in his party have expressed doubts about whether to override Obamacare before agreeing to a new plan.
Others have been in favor of revoking it, but keeping it provisionally enforceable while negotiating another system.
A priori, Republicans have enough votes to override the program, but they would need the support of some Democrats to push through a new law.
In his interview with The New York Times, Trump warned Democrats about the risk of trying to oppose change, especially senators from some states who have to opt for re-election in 2018.
"I won some of those states with numbers that no one had seen. He will be there campaigning (against them), "he said.
The so-called "Obamacare" was one of the great stakes of the Barack Obama Administration and has provided health insurance to millions of Americans.
The plan, however, has always been opposed by the Republicans and Trump promised during his campaign that he would end it.
Since his election victory, Trump's team has made it clear on several occasions that as soon as he arrives at the White House he will use his executive powers to begin dismantling the law.
According to the spokesman of the Presidential Transition Team Sean Spicer, Trump had several hours left from his day to discuss health issues with his most direct advisors.
With information from AFP