Justice Samuel Alito writes that the 1973 Roe v Wade decision – that legalised abortion across the US – is “egregiously wrong” in a 98-page draft opinion.
If the ruling comes to fruition, "trigger laws" could instantly make abortion illegal in 22 US states.
Such a ruling is not expected until early July.
The court is weighing up a challenge to Mississippi's ban on abortion after 15 weeks, which the justices heard in December.
The reported threat to Roe v Wade comes at a time when reproductive rights are being threatened in Republican-leaning states across the US.
Protestors react outside the US Supreme Court
MOIRA WARBURTON
Protesters react outside the US Supreme Court to the leak of a draft majority opinion written by Justice Samuel Alito
JONATHAN ERNST
It sparked immediate outcry from Democrats and protests by both pro and anti-abortion supporters.
Speaker of the House of Representatives Nancy Pelosi and Senate Majority Leader Chuck Schumer issued a joint statement saying that if the report was accurate, the "Supreme Court is poised to inflict the greatest restriction of rights in the past 50 years".
And it sparked international condemnation, too.
In the UK, International Trade Secretary Anne-Marie Trevelyan said the overturning of Roe v Wade will cause a "great deal of anxiety" among women in the US.
Protesters reference Margaret Atwood's dystopian novel The Handmaid's Tale in Washington
MOIRA WARBURTON
News of the leak has sparked outrage in the US
MOIRA WARBURTON
The report, published by Politico, quoted Justice Alito as saying: "Roe was egregiously wrong from the start.
"Its reasoning was exceptionally weak, and the decision has had damaging consequences.
"And far from bringing about a national settlement of the abortion issue, Roe and Casey have enflamed debate and deepened division."
Neither the Supreme Court nor the White House commented on the leak.
Six of the nine justices currently on the court were appointed by Republican presidents, with the other three picked by Democrats.
Politico reports that Justice Alito and four other Republican-appointed justices – Clarence Thomas, Neil Gorsuch, Brett Kavanaugh and Amy Coney Barrett – were in favour of the move but it is not clear how Chief Justice John Roberts will vote.
The leaked document is marked as a "1st Draft" and was reportedly circulated in February.
The American Civil Liberties Union said that if confirmed, the decision "would deprive half the nation of a fundamental, constitutional right that has been enjoyed by millions of women for over 50 years".