The Italian foreign minister has demanded details “in the next few hours”, an Italian news agency said, after accusations the UK did not inform Italy about the planned mission.
Islamist militants took British Chris McManus, 28, of Oldham, and Italian Franco Lamolinara, 48, hostage in north-west Nigeria last May.
The engineers died as Nigerian and UK forces tried to free them on Thursday.
UK Foreign Secretary William Hague and Giulio Terzi are attending a meeting in Denmark.
William Hague earlier said that the limited time available to make a decision “constrained how much we were able to consult others”.
Giorgio Napolitano said the UK needed to explain why it did not inform the Italian authorities ahead of the operation.
“The way the British government has behaved is quite inexplicable. To have failed to inform or consult Italy, with regard to a military action which could have such consequences,” the Italian president said.
“A clarification is needed on both the political and diplomatic levels.”
UK Defence Secretary Philip Hammond said Rome had been told of intelligence behind the rescue attempt and informed “as the decision was taken to act”.
Philip Hammond said that the attempted rescue had been launched after information was received that Chris McManus and Franco Lamolinara “were about to be moved – possibly executed”.
He said what had subsequently happened was “very unfortunate but it’s completely explicable”.
“These hostages were taken, they were held at an unknown location for a very long period of time despite extensive efforts to track them down. And when a window of opportunity became available, a well-trained Nigerian force with British support, went in and tried to rescue them,” Philip Hammond said.
“The Italian government was kept informed throughout the operation as the intelligence emerged and then as the decision was taken to act… I don’t think they specifically approved it – they were informed of what was happening.”
Number 10 said earlier that no official complaint had been received from the Italian government, and the UK had not made an apology.
It said the UK and Italy had been in contact ever since the men were kidnapped on 12 May, 2011.
Prime Minister David Cameron’s official spokesman said: “We contacted the Italians yesterday as the operation was getting under way, but this was a very fast-moving situation.
“Our priority was to respond to the situation on the ground and to do everything we could to try and secure the safe release of the hostages.”
In a statement announcing the operation and the deaths of Chris McManus and Franco Lamolinara, PM David Cameron earlier said the decision to act had been taken at very short notice.
“A window of opportunity arose to try and secure their release. We also had reason to believe that their lives were under imminent and growing danger.”
It appeared that the Italian government tended to be in favor of negotiating in hostage situations, while Britain was absolutely against formal government negotiations.
News of the operation broke in a statement from the office of Italian Prime Minister Mario Monti.
It used very similar language to that used by the British side but made it clear the Italians were told the military was being used only after the operation had started.
Mario Monti chaired a meeting of a government security committee to discuss the failure of the attempt to rescue the hostages. The meeting ended after two hours without any comment being issued.
Commons Foreign Affairs Select Committee chairman Richard Ottaway said the UK was not duty-bound to tell Italy about the operation in the circumstances.
Richard Ottaway said: “I can understand the concerns and frustrations of Italian politicians but I think they’ve got to accept and recognize that these are very fast-moving, delicate operations and it’s not always possible to keep politicians briefed in advance of what goes on.”
David Cameron’s spokesman said it was a Nigerian-led operation, with UK support involving the UK’s elite Special Boat Service (SBS).
The house where the kidnapped men were being held had been under surveillance for some time.
David Cameron and Goodluck Jonathan said they believed the kidnappers had killed Chris McManus and Franco Lamolinara.
But an unnamed official from the Nigerian state security service quoted in local reports said the hostages died in the crossfire.
The Nigerian president described it as a “deeply sad and regrettable incident”.
Goodluck Jonathan said that the men’s captors had been seized and “would be made to face the full wrath of the law”.
The Nigerian president said they were from militant Islamist group Boko Haram, which has carried out a number of attacks on police, politicians and clerics who oppose it.
Reports have emerged that a senior member of Boko Haram was captured on Tuesday, and he gave information which led forces to the house where the two construction engineers were being held.
However, on Friday, a spokesman for the group told reporters Boko Haram was not responsible for the deaths.
“We have never taken anyone hostage. We always claim responsibility for our acts,” he said.
Boko Haram has become more violent and capable in recent years and there is a suspicion this may be a sign of the growing influence of the group al-Qaeda in the Islamic Maghreb.
Al-Qaeda in the Islamic Maghreb has a track record of kidnapping Westerners in North Africa and trying to extract ransom payments to fund its violence.
British Chief of Defence Staff General Sir David Richards said that the rescue attempt had sent “a shockwave” through al-Qaeda in the Islamic Maghreb.
“We know that all their leaders are now and disarray and we’re now obviously trying to track them down with our Nigerian allies,” David Richards said.
Gunmen seized Chris McManus and Franco Lamolinara in the town of Birnin Kebbi in the north of Nigeria on 12 May 2011.
Both men worked for Italian firm B Stabilini in the construction of a local headquarters for the Central Bank of Nigeria.
Relatives of Chris McManus released a statement saying they were “devastated” by his death but thanked those who had worked to try to free him.
“During this ordeal we have relied heavily on the support of our family and friends which has never waned and has enabled us to get through the most difficult of times,” the family statement said.
The UK Foreign Office advises against all travel to some areas of Nigeria and against all but essential travel to other areas, but these do not include the area where the men were kidnapped.
It says there is the threat of kidnap and terrorism across the West African country.
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