Martin Nesirky, Ban’s spokesman, said the UN chief was "deeply disturbed by the reported deaths" of the prisoners as they were being transferred to a different facility in Cairo, Al Jazeera reported.
"He calls for a full investigation to ascertain the facts surrounding this incident," Nesirky said.
Authorities said 37 detainees died after police fired tear gas in a bid to free an officer taken hostage by prisoners, as the inmates were being transferred to Abu Zaabal prison near Cairo.
But the Anti-Coup Alliance, which includes the Muslim Brotherhood, held the police accountable, accusing them of "murder".
The Alliance has demanded a formal investigation into the deaths in disputed circumstances on Sunday.
Divergent explanations have emerged for the deaths of the prisoners.
Photos provided by the lawyers representing the detainees show dead bodies with charred faces and limbs and others covered in bruises which the lawyers said were signs of torture.
The U.S. State Department termed the deaths "suspicious".
"We are ... deeply troubled by the suspicious deaths of Muslim Brotherhood prisoners in a purported prison escape attempt near Cairo," State Department spokeswoman Jen Psaki told reporters.
Arrests made
Having initially defended police action, Egypt later said that two officers had been arrested over the incident.
Egypt's second largest Islamic party, Nour, called in a statement for a presidential decree to create an independent fact-finding committee into the deaths and blamed the interior ministry for the incident.
Amid government crackdown on Brotherhood, the UN chief has said "the political space for the Muslim Brotherhood should be expanded, because their political space has been very limited".
The US has said that it was not in favor of banning of Muslim Brotherhood after the military-installed government indicated they were planning to ban the Islamist group.
European Union Foreign Ministers will hold an emergency meeting in Brussels on Wednesday to discuss how to force Egypt's army-backed government into seeking a peaceful compromise.
Options likely to be discussed include cutbacks in Europe's $6.7bn package of grants and loans promised last year, as well as a possible arms embargo against Egypt, said EU envoy Bernardino Leon.
Saudi Arabia pledged to fill any financial gaps left by Western countries. Israel, fearful of unrest, is also prodding the West to stick by Egypt's army.
On Monday, Egypt's public prosecutor ordered the deposed president, Mohamed Morsi, to be detained for 15 days over allegations he participated in "violent acts" last December. The 15-day order will begin at the end of a 30-day order issued on Thursday related to allegations of collusion with foreign groups and spying.
About 900 people, including nearly 100 soldiers and police, have died in Egypt since the authorities on Wednesday began to forcibly break up Muslim Brotherhood sit-ins by supporters of Morsi who was toppled in a July 3 military coup.
MNA
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