Yakub Memon says the authorities did not follow proper legal procedures in issuing the warrant for his death.
On Tuesday the court rejected a plea for mercy – paving the way for his execution, possibly later this month.
The serial blasts killed 257 people and were allegedly to avenge the killing of Muslims in riots a few months earlier.
Another 713 were wounded in the bomb blasts in India’s financial capital, then called Bombay.
Reports quoting Memon’s lawyer said his death warrant was issued before the Supreme Court dismissed the mercy plea.
India rarely sentences people to death – and even more rarely carries out executions. There have been three since 1995.
If the top court rejects Memon’s appeal, he will be the first to be executed since a Kashmiri man, Afzal Guru, was hanged in 2013 for the 2001 attack on parliament.
Mohammed Ajmal Qasab, the sole surviving gunman from the 2008 Mumbai attacks, was executed in November 2012.
In 2004 a Kolkata (also called Calcutta) security guard was hanged for the rape and murder of a schoolgirl.
Memon, a chartered accountant, was sentenced to death in 2007 by a special court in Mumbai, which found he played a key role in the bombing conspiracy.
He is being held in a prison in the western city of Nagpur.
A total of eight members of the Memon family were initially accused of masterminding the bombings and dispersing funds for the attacks.
The eldest brother fled the country, and three other family members were acquitted for lack of evidence.
The alleged masterminds of the blasts, Dawood Ibrahim and Tiger Memon, have been on the run since 1993.