“We tell the countries, which are taking part in the Crusade campaign: By God’s command, you will have your day that will be similar to the day of France,” a fighter in the purported video says.
“I swear by God, we will pound America on its land. We will conquer Rome.”
Last year, the US launched a multinational aerial campaign against the radical group in Iraq and neighbouring Syria.
Islamic State, which rules vast territory in both countries, has claimed responsibility for Friday’s terrorist attacks in Paris that killed at least 129 people.
The 11-minute video features several alleged fighters from Islamic State.
“We tell the countries of Europe we are coming with car bombs and weapons. You won’t be able to make us retreat,” says a man in the video.
The authenticity of the recording could not be independently verified.
In the aftermath of the Paris attacks, France vowed to step up its fight against Islamic State.
On Sunday, 10 French jets pounded the group’s de facto capital of al-Raqqa in northern Syria, hitting training camps and weaponry depots, activists said.
Rami Abdel-Rahman, the head of the Britain-based Syrian Observatory for Human Rights, reported that in total there were 36 strikes targeting Islamic State in the area.
He told dpa that in recent weeks Islamic State, sensing growing pressure on its key territory, had begun to move families of foreign jihadi fighters out of al-Raqqa city.
“They were moved to areas under the Islamic State’s control in other parts of Syria … which the movement thinks is safer,” Abdel-Rahman said, adding that troops were also redeployed to new positions in the area.
The Raqqa Revolutionary Brigades, a Syrian rebel movement allied with the Kurdish Peoples’ Protection Units (YPG), meanwhile announced it has graduated a new batch of fighters, in a video posted on YouTube.
The combined forces have been indicating in recent weeks they are planning to move on Islamic State territory in al-Raqqa, south of the rebels’ current positions.
YPG and Arab forces were the main group on the ground defeating Islamic State this year, securing victories in Kobane and Tel Abyad, along the Syrian-Turkish border, and retaking the cities from the extremists.
An activist with family in al-Raqqa told dpa that people in the city were being told to relocate.
“Civilians who are living next to key bases for the movement like the religion police and other key bases were asked to move out,” the activist said by telephone.
He noted that the situation inside al-Raqqa is deteriorating, amid ongoing airstrikes by the US-led coalition and Russia.
Al-Raqqa was first taken in 2013 by hardline Islamist rebels, including al-Qaeda-linked al-Nusra Front, ousting forces loyal to Syrian President Bashar al-Assad.
Later that year, Islamic State dislodged the rebels and seized the city for itself.
Islamic State has since imposed strict Islamic law on the city, including corporal punishment for moral violations and restrictions on the Christian minority.
Kurdish forces in northern Syria said the Turkish military opened fire on its position near Kobane, killing one civilian.
The incident took place on Friday night near Siftek village, to the west of Kobane.
The YPG said in a statement that Turkish forces also fired Sunday night at its position near Tel Abyad.
There was no official Turkish comment.