Relatives will mourn in private.
"The family wants to avoid a media circus," an unidentified person told the New York Daily News.
The body was picked up from the New York City morgue on Tuesday and was set to be cremated later that day, The New York Times and The Wall Street Journal reported.
Mark Madoff was said to be depressed about an ongoing criminal investigation to determine if other Madoff family members knew about the scam, and by lawsuits being filed by a court-appointed trustee trying to recover money for victimized investors.
Madoff and his brother, Andrew, worked at their father's firm, alerting authorities to the scam in 2008 and denying any knowledge of the crimes.
"Mark was an innocent victim of his father's monstrous crime who succumbed to two years of unrelenting pressure from false accusations and innuendo," Martin Flumenbaum, Madoff's lawyer, said in a statement.
Bernard Madoff, serving a 150-year prison sentence for the largest U.S. investment fraud in history, said earlier through his lawyer he would not seek permission to leave prison in North Carolina to attend a service. He said he, too, would mourn alone.
His wife, Ruth Madoff, reportedly told acquaintances this week she was furious with her husband, blaming him for Mark's death.