Theresa May’s record as Home Secretary has been thrown into doubt over claims she weakened surveillance on an extremist who fled the country to become an Isis fighter and suicide bomber after his release from Guantanamo Bay.

The Prime Minister’s official spokesman refused to discuss whether monitoring of Jamal al-Harith had stopped in 2014, at which point Ms May had been running the Home Office for four years.

But two ex-Labour home secretaries, David Blunkett and Jack Straw, attacked Ms May’s handling of terror suspects, including Harith.

The 50-year-old suicide bomber is said to have received £1m, while Ms May was Home Secretary, from the Government in compensation for his detention at Guantanamo Bay. Isis has claimed Harith, born Ronald Fiddler in Manchester, blew himself up in an explosives-laden vehicle in a village south of Mosul.

The terror group said there had been many casualties, though this has not been officially confirmed, and released a picture of a grinning Harith, who also went by the name Abu Zakariya al-Britani.

Former Prime Minister Tony Blair yesterday took the unusual step of issuing a statement denying his administration had paid compensation to Harith, a claim made by a national newspaper, and pointing out that the money was handed over in 2010 under a Conservative government.

With focus shifting on to Ms May’s tenure at the Home Office, her spokesman refused to answer questions on the matter at two separate briefings.

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No 10 declined to comment on the compensation payments, whether they were necessary, if Ms May had agreed with them or tried to stop them. The spokesman also refused to say why the Prime Minister as Home Secretary had allowed Harith to travel to Syria.

Ms May’s spokesman said to each question: “It is an intelligence matter.”

He later added: “It is being reported as a matter of fact about this man, but there is no independent confirmation of the identity of this man who is believed to be dead in Mosul.”

Lord Blunkett suggested in a statement that monitoring of the released Guantanamo detainees had stopped when Ms May was Home Secretary.

He said: “I am not aware as to the length of time such monitoring continued after I left the Home Office at the end of 2004.

“It is clear however that in 2010 under the new Coalition government, when the compensation awards were actually given, continuing contact and awareness of these individuals must have been present.

“What happened between then and 2014 is of course a matter of speculation that can only be answered by the present Government.”

Meanwhile Jack Straw followed up the attack, telling The Independent: “One of the things that she [the Prime Minister] did do was weaken the ability of Government to keep terror suspects under effective surveillance.”

He pointed out that Ms May had abolished control orders introduced by Labour and replaced them with orders under the Terrorism Prevention and Investigation Measures Act, known as TPIMs.

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While security officials confirmed that Harith and other Guantanamo detainees had been kept under surveillance after their release, they noted that the scale of it depended on the extent of the threat they posed to the public.

One senior source said: “There is a finite amount of resources and a valued judgement has to be made on how these resources are allocated.

“As far as I am aware there was no indication that he was involved in a terrorist plot in this country.”

Richard Barrett, the former head of counter-terrorism at MI6, said: “This guy came out of Guantanamo Bay ten years before he left for Syria. That would be a hell of a long time to keep someone under close surveillance, almost impossible, and against his civil liberties.

“When he came back from Guantanamo Bay the assessment was made by the British and Australian services that he posed no threat and for the ten years that he was in the UK and maybe even for a period after that, he did not pose any direct threat to the British public in the way that we would understand it.”

Jamal al-Harith, a former inmate at Guantanamo Bay, gestures after his testimony to Europe's top human rights body in Paris in 2004 (AP)

Mr Blair decided to make a statement on the matter after a news story in which a Tory MP attacked his “flawed judgement” over Harith, in particular citing the compensation paid to him.

Harith’s family have since denied that he received £1m in compensation, claiming the figure was “a group settlement including costs for four innocent people including Jamal”.

The Manchester man was seized by American forces in Pakistan in 2001 before being sent to Guantanamo Bay, the US prison in Cuba for terrorist suspects.