Doctors linked to the opposition said the bodies were among 100 people believed killed since security forces attacked a protest camp on Monday.
Reports said a feared paramilitary group was attacking civilians.
Sudan’s ruling Transitional Military Council (TMC) vowed to investigate.
Residents in Khartoum told the BBC they were living in fear as members of the Rapid Support Forces (RSF) roamed the streets. The paramilitary unit – formerly known as the Janjaweed militia – gained notoriety in the Darfur conflict in western Sudan in 2003.
“Forty bodies of our noble martyrs were recovered from the river Nile yesterday,” the Central Committee of Sudanese Doctors said in a Facebook post.
BBC has also reported that they had witnessed and verified the bodies in hospitals and that the death toll now stood at 100.
A former security officer quoted by Channel 4’s Sudanese journalist Yousra Elbagir said that some of those thrown into the Nile had been beaten or shot to death and others hacked to death with machetes.
“It was a massacre,” the unnamed source said.
On Wednesday, the head of Sudan’s Transitional Military Council (TMC), General Abdel Fattah al-Burhan, apologised for the loss of life and called for resumed negotiations – reversing a statement the previous day in which he said dialogue was over.
But a Sudanese alliance of protestors and opposition groups rejected the invitation. One of its leading members said the TMC could not be trusted.
The deputy head of the TMC defended the violent suppression, claiming that the protesters had been infiltrated by rogue elements and drug dealers.
“We will not allow chaos, we will not allow chaos and we will not go back on our convictions. There is no way back. We must impose the respect of the country by law,” said Mohammed Hamadan – also known as Hemedti