Aljazeera reports quoting Iraq President Jalal Talabani as saying on Wednesday that the 50 bodies found near al-Suwaira were those of people recently abducted from al-Madain town, south of Baghdad.
Agencies quoted him as telling reporters: "More than 50 bodies have been brought out from the Tigris and we have the full names of those who were killed and those criminals who committed these crimes.
"We will give you details in the coming days … terrorists committed crimes there. It is not true that there were no hostages. There were, but they were killed and they threw the bodies into the Tigris."
Shia officials said on Saturday that about 50 people had been taken hostage by Sunni fighters in al-Madain and were threatened with death.
Iraqi security forces raided the town but said they found next to no evidence that anyone had been taken hostage or that any armed men were in the town.
Later, Shia officials said dozens of bodies had been found in the Tigris south of al-Madain, but residents and police in the area who spoke to Reuters said they had not seen any bodies.
Separately, the bodies of 19 Iraqis were left in a football stadium in Haditha, a town 220km northwest of Baghdad, an Iraqi reporter and residents said.
Iraqi police and army personnel
are being attacked regularly
An Interior Ministry official said the victims were Iraqi soldiers executed by anti-government fighters on Wednesday.
The reporter and other residents counted 19 bodies and said all of them appeared to have been shot.
Residents said they thought the victims - all men in civilian clothes - were soldiers abducted by anti-US and anti-government fighters as they headed home for Thursday's holiday marking the birthday of the Prophet Muhammad.
The reporter did not see any military identification documents on the bodies and it was not immediately possible to confirm the claim.
Addressing the media, Talabani also said the Prime Minister Ibrahim al-Jafari would unveil the new Iraqi cabinet on Thursday, Aljazeera reports.
"We want to announce it as soon as possible," he said after the meeting on Wednesday with al-Jafari, former interim prime minister Iyad Allawi and Abd al-Aziz al-Hakim, the leader of the Shia political party Supreme Council of Islamic Revolution in Iraq (Sciri).
Iraqi leaders have been negotiating over the formation of the government since the 30 January elections that brought a Shia Muslim majority to power.
News of the alleged abductions
ratcheted up sectarian tension
But disagreements over which parties should get which ministries, and on how the Sunni minority should be brought into political process, have held up the formation of the government.
Al-Jafari, the leader of one of two main Shia parties in the coalition that won the election, was formally appointed prime minister on 7 April and in theory has a month from that date to announce the make-up of his cabinet.
Much of the squabbling has focused on the Oil, Interior and Defence Ministries. The Interior Ministry is expected to go to a member of Sciri, the main party in the Shia alliance.