The conservative federal government has challenged the law, backed by anti-abortion groups and the Roman Catholic Church in the devoutly religious country.
The votes of eight out of 11 supreme court judges are necessary for the Mexico City law to be abolished, and four of the judges have not yet publicly revealed how they will vote, including the court's only two female judges.
Since April 27 last year, when the abortion law was first applied, 12,262 women aged between 18 and 29 have had abortions in one of 12 clinics.
"That means that some 80 women per day have exercised their controversial right," Maria Luz Estrada, spokeswoman for the Catholics for the Right to Decide organization told AFP.
Some 14,000 women decided not to abort after receiving information about abortions, Estrada added.
In the rest of Mexico, states allow abortions only under limited circumstances, such as rape and incest, but rights groups say that in practice such abortions are difficult to obtain.
If the high court abolishes the Mexico City law, women who have abortions will face prison sentences of between three and six months.
Mexico City lawmakers organized a demonstration Monday at one of the city's largest women's prisons to protest the possible law reversal.
Between 1990 and 2005, an average of 13 women died per year due to clandestine abortions in Mexico City, according to pro-abortion groups. The ultra-conservative Provida organization said eight women had died per year.
Since last April, only one 16-year-old has died during an abortion, due to her doctor being misinformed about the length of her pregnancy.
Provida on Sunday attached 12,000 paper crosses on the steps of Mexico City's Angel of Independence monument to mark the number of abortions carried out since the law was passed last year.
The Mexican Bishops' Conference has also spoken out against the law.
The court is expected to vote later this week.