ISIS claimed responsibility for the Charlie Hebdo shooting in which 11 people were killed in Paris in January this year.
The group have also recently claimed they are responsible for the downing of a Russian Airliner which crashed in Egypt, killing everyone on board.
The attacks in Paris come after yesterday's U.S. drone strike in Syria which targeted the masked Islamic State militant known as "Jihadi John".
U.S. military spokesman Steve Warren said officials were "reasonably certain" the strike killed the British man who appears in several videos depicting the beheadings of Western hostages.
Twitter accounts linked to jihadists are celebrating the attacks in Paris.
According to the SITE Intelligence Group tracking militant sites, Twitter posts attributed to jihadist supporters are speculating which group may be responsible. Many users expressed belief that the Islamic Group could be behind the carnage.
They used Arabic-language hashtags that translated to "Paris on fire" - the same hashtag ISIS used to celebrate the Charlie Hebdo terror attack - and "Caliphate state strikes France."
SITE says that accounts also circulated pictures of the attacks, and one pro-IS channel accused France of sending warplanes to bomb Syria and says "today it drinks from the same cup."
James Woolsey, a former director of the CIA in 1993-195 and now chancellor at the Institute of World Politics, also told the BBC he suspected the Islamic State because the coordinated nature of the attacks required government-style planning.
There has been no official comment on motive.
The shooting of hostages will have forced the French government's hand with the prospect of the seige in Paris ending with a large loss of life, says intelligence analyst Paul Buchanan.
The former adviser to the CIA spoke as the horror of the attacks unfolded in Paris with as many as 60 people killed and a mass taking of hostages by a reported six gunmen. There were reports of some hostages escaping and others being shot one by one.
"You can't control 100 hostages with six gunmen," said Buchanan, whose 36th Parallel consultancy provides intelligence and security advice. "Those that escape will be able to provide intelligence on who the attackers are."
He said the method of attack adopted by the attackers raised questions about the group they represented.
He said the use of bombs and gunfire was a blend of tactics usually employed separately by Al Qaeda and ISIS. "Normally they do one or the other."
He said it could be that the attackers were local with sympathetic to ISIS or north African.
"You're going to lose a lot of lives. If they keep shooting people every five minutes then the government's hand is forced."
Reporting by David Fisher of NZ Herald and AP