The document, published by Politico, suggests the country's top court is poised to overturn the 1973 decision that legalised abortion nationwide.
If the court strikes down the Roe v Wade ruling, individual states would be allowed to ban abortion, if they wish.
It is expected abortion could then be banned in almost half of US states.
The Supreme Court's justices are expected to issue a ruling in late June or early July.
Roe v Wade is in the court's sights because Mississippi is asking for it to be overturned. The justices heard that case in December.
Some 36 million women could then lose abortion access, according to research from Planned Parenthood, a healthcare organization that provides abortions.
Anti-abortion groups such as the Susan B. Anthony List welcomed the news.
If Roe is indeed overturned, our job will be to build consensus for the strongest protections possible for unborn children," it said.
The leaked document - labelled "1st Draft" - appears to reflect the majority opinion of the court, and Politico reports that it was written by Justice Samuel Alito and circulated within the court on 10 February