Calling for an international coalition to confront the insurgency on its border, Egypt confirmed it had conducted raids in the IS-stronghold of Derna, in Libya's east, backed by Libya's air force.

Libya's Prime Minister Abdullah al-Thinni also issued a plea for Western military intervention as the country's already dire security situation slipped further into chaos.

"We have absolutely confirmed information that al-Qaeda and IS are in Tripoli and . . . near Ben Jawad" on the country's central coastline, he said.

"I ask world powers stand by Libya and launch military strikes against these groups," he said. "This threat will move to European countries, especially Italy."

At least 64 militants were killed in Egypt's attacks, Libyan Air Force commander Saqer al-Joroushi told Egyptian state television, indicating further air strikes were planned for Tuesday.

Local residents say civilians also died in the air strikes, although it was not possible to confirm these casualties before publication.

The Coptic labourers were kidnapped by Islamic State militants in Sirt, on Libya's coast, in two attacks in December and January.

In what has become a hideously familiar pattern, the slickly-produced IS video shows the hostages, dressed in orange jumpsuits, being simultaneously beheaded by their individual executioners.

This time, the video ends with both a broad threat to the West and a specific threat to Italy, which lies just across the Mediterranean from Libya.

The knife-wielding spokesman points north after the executions and says: "We will conquer Rome, by Allah's permission."

Italy says it is already deeply concerned about the growing numbers of desperate asylum seekers attempting to cross into Italy from the coast of Libya – more than 2000 were rescued just this weekend alone.

Describing the Islamic State as just "350 kilometres from our shore," Defence Minister Roberta Pinotti told the newspaper Il Messaggero that Italy was ready to send more than 5000 soldiers under a United Nations mandate to stop IS.

"We've been discussing it for months, but now the need for intervention has become urgent."

The brutal deaths of the Egyptian Christians comes just months after a group of Libyan jihadists made their October pledge of allegiance to IS. By November the self-proclaimed caliph, Abu Bakr al-Baghdadi, confirmed the expansion of IS into Libya in a recording released online.

Engulfed in a crisis that has crippled the country's parliament and economy and fuelled the growth of rival governments, armies and militia, Libya is now one of IS's fastest-growing new franchises.

Up to 3000 people have been killed in Libya in the last 14 months as rival groups battle across multiple fronts for territory, legitimacy and power, creating perfect conditions for jihadist groups like IS to gain a foothold.

For Egypt, already fighting a terrorist insurgency in north Sinai against an IS-affiliated militia, the dangers of being drawn into the rapidly deteriorating security situation in Libya are enormous, analysts say.

If Egypt progresses from air strikes to troop deployment along its border with Libya, it would be opening a second front in its fight against terrorism, overstretching its military and drawing resources away from Sinai.

"[IS] will have hoped to provoke [Egyptian] airstrikes in Libya," Charles Lister, a visiting fellow at the Brookings Doha Centre whose research focuses on IS tweeted on Monday. "This fits perfectly its vision of escalating conflict and chaos in the region."

Others warned that Egyptian military intervention could undermine both Egypt and Libya's prospects for stability. "The strength of the Islamists in the east should not be underestimated," say Karim Mezran and Tarek Radwan from the Atlantic Council's Rafik Hariri Centre for the Middle East.

"They are very capable of holding their territory and could begin heavy guerrilla warfare against what at that point could easily and justifiably be portrayed as foreign 'invaders'."

This could push even more Libyans to throw their support behind radical groups such as IS, the two wrote in the Cairo Review last month.

Egypt's President Abdel Fattah al-Sisi, who has waged his own ferocious security crackdown against the Islamist Muslim Brotherhood, described the deaths of the 21 Copts as an "abhorrent act of terrorism".

He warned: "Egypt and the entire world face a fierce battle against terrorist organisations that are bound by the same extremist thought.

"We must therefore stand united to root it out and protect the world from the proliferation of this toxic menace."

At the heart of Libya's meltdown, which began soon after its 2011 revolution forced the overthrow of dictator Muammar Gaddafi (he was later killed by an angry mob), is its competing governments and the militia who back them.

The official, internationally-recognised government moved from the capital Tripoli to Tobruk in August last year after a spate of kidnappings, arrests and killings drove politicians to seek a safer haven in the east.

The group behind that violence, Libya Dawn, is a loose coalition of Islamist militias including former al-Qaeda jihadists, tribal militia and some of the country's Muslim Brotherhood members.

Despite a requirement that they step down after the June 2014 election, members of the previous Libyan parliament - dominated by supporters of Libya Dawn - continues to sit in the capital Tripoli.

And while the government in Tobruk supports Egypt's military intervention, the unofficial government in Tripoli does not, describing it as an assault on Libya's sovereignty.

The forces controlled by General Khalifa Haftar, a former Gaddafi-era officer, are backing the official government in Tobruk and doing battle against Libya Dawn across the country. His fighters are mainly drawn from those seeking greater autonomy for the eastern region of Cyrenaica, as well as tribal fighters and former military officers.

The Libyan National Army, as General Haftar calls it, is financed by Egypt, Saudi Arabia and the United Arab Emirates, analysts say, while Libya Dawn is backed by Qatar and Turkey.

With at least 100,000 more Egyptians believed to be in Libya – like those who were murdered, they are labourers seeking work outside Egypt's decimated economy – the government has also committed to helping them return to Egypt.

There are already reports that at least 35 Egyptians have been kidnapped in areas controlled by Ansar al-Sharia and IS, local media reported overnight.

In the meantime, Egypt's Minister of Foreign Affairs has travelled to New York to hold talks on Libya with the United Nations Security Council.