Mr Henning was delivering aid to Syria when he was kidnapped, as Paul Wood reports
The Salford taxi driver was delivering aid to Syria in December when he was kidnapped and then held hostage by IS.
IS threatened to kill him in footage last month showing the death of Briton David Haines, and in this video they threaten US aid worker Peter Kassig.
David Cameron said Britain would do all it could “to hunt down these murderers and bring them to justice”.
The prime minister said the killing of father-of-two Mr Henning, 47, showed “how barbaric and repulsive” IS was.
“My thoughts and prayers tonight are with Alan’s wife Barbara, their children and all those who loved him,” he said.
“Alan had gone to Syria to help get aid to people of all faiths in their hour of need.”
Mr Henning’s wife Barbara had this week appealed for her husband’s release, saying: “He is innocent.”
Volunteer Mr Henning was on his fourth aid mission to Syria when he was captured within minutes of arriving in the country last December.
The prime minister will be briefed by intelligence and security chiefs on Saturday.
Downing Street said the “barbarity” of the act “underlines why it is right for Britain to join in the attacks against IS”, according to BBC assistant political editor Norman Smith.
“It is also pointed out that ministers have known for some time the risk to Western hostages and cannot allow that to determine British foreign policy,” our correspondent added.
Number Ten has declined to comment on the possible use of special forces in the fight against IS.
IS has previously released videos showing the apparent beheading of two US journalists, James Foley and Steven Sotloff, and British aid worker Mr Haines.
The video released on Friday is yet to be verified, but it appears to show Mr Henning kneeling beside a militant, who is dressed in black, in a desert setting.
The footage ends with an IS fighter threatening a man they identify as Mr Kassig.
In a statement, Mr Kassig’s family said he had converted to Islam and referred to him as Abdul Rahman Kassig.
The family asked people around the world to pray for his release and that of “all innocent people being held hostage in the Middle East and around the globe”.
They also asked people to pray for Mr Henning’s family, adding: “We have read about his work and his generous character with great respect and admiration.”
Mr Henning’s friend Majid Freeman described him as a “selfless, humble, courageous individual” who had simply wanted to help others.
Mr Freeman, who was with him on the convoy when he was captured, told BBC Radio 4’s Today programme: “He was helping the innocent people the rest of the world had abandoned.
“It doesn’t make sense to kill him.”
BBC security correspondent Gordon Corera said the footage was similar to previous videos released by IS, though slightly shorter.
He said it included a reference to last week’s vote by UK Parliament to authorise air strikes against IS in Iraq.
Like previous videos it features a militant with an apparently British accent, he added.
The UK Foreign Office said in a statement: “We are aware of the video and are working urgently to verify the contents.
“If true, this is a further disgusting murder.
“We are offering the family every support possible; they ask to be left alone at this time.”
The US said it was evaluating the video and if proved real, it was “another demonstration of the brutality” of the militant group.
Frank Gardner, security correspondent, BBC News
David Cameron’s vow to catch the jihadist killers of Alan Henning and bring them to justice would seem to match the mood of the nation.
But judging by the track record of previous such cases of hostages being murdered overseas, this promise has little likelihood of being fulfilled.
Tony Blair made the same pledge after Ken Bigley from Liverpool was beheaded in 2004, Gordon Brown did the same when tourist Edwin Dwyer was kidnapped and killed in the Sahara, and Mr Cameron vowed to punish those who besieged the Algerian gas plant last year.
According to the Crown Prosecution Service, there has not been one single case of any murderers of British hostages ever being brought to justice in Britain.