Instead, Theresa May will attempt to convince a British high court judge that she has made sufficient "demonstrable progress" in securing assurances from the Jordanian government about Qatada's treatment to prevent him being freed from virtual house arrest.
The announcement is expected to lead to Qatada's lawyers making an application to the special immigration appeals commission for the highly restrictive bail conditions on the terror suspect to be lifted.
May's Commons statement comes after the deadline for lodging an appeal to the Strasbourg court passed at midnight on Monday. May is trying to secure assurances from the Jordanians that Qatada will not face a trial on his return based on evidence obtained by torture, but has yet to reach a final agreement.
Qatada, 52, whom a Spanish judge once described as Osama bin Laden's right-hand man in Europe, is on bail under 22-hour curfew pending the outcome of May's negotiations with Jordan.
The case has proved an embarrassment to the home secretary and fuelled government efforts to reform the Strasbourg court. Ken Clarke, the justice secretary, will chair a Council of Europe conference in Brighton this week.
The decision not to appeal is believed to be based on the fact that the ruling upheld the general right of the British government to try to deport international terror suspects with diplomatic assurances about their future treatment. Home Office officials feared that this important principle could be lost if an appeal to the Strasbourg court's grand chamber was defeated. Several other deportation cases could have been put at risk if an appeal failed.
May's statement is likely to trigger an application by Qatada's lawyer, Gareth Peirce, to lift bail restrictions. At an expedited hearing, possibly as early as this week, the special immigration appeals commission judge, Mr Justice Mitting, will decide whether to free Qatada completely or return him to Long Lartin maximum-security prison pending a new effort to deport him.
The midnight deadline marked three months since the Strasbourg judges' January ruling that Qatada could not be sent back to Jordan because of the serious risk that he would face a trial based on evidence obtained by torture. Mitting also gave the home secretary three months to demonstrate progress with the Jordanians when he announced his decision on 6 February to release Qatada on bail.
May has been keen to stress the "positive progress" made since she visited Jordan last month seeking assurances that torture-tainted evidence would not be used in Qatada's case. British officials visited Jordan again two weeks ago and discussions were said to be down to the "nitty gritty legal detail". The difficulty remains that although Jordan says it has banned the use of torture, the Strasbourg judges ruled that its use remains widespread and routine.
Qatada has spent more than six and a half years in detention in maximum-security prison or under 22-hour curfew in Britain since he was declared a threat to national security as a terror suspect.
The shadow home secretary, Yvette Cooper, said it was time May got a grip of the case. She said: "The home secretary and the British courts have made clear that Abu Qatada is a serious threat to national security. Yet he continues to be allowed on Britain's streets and we now face the possibility of an appeal to relax his bail conditions. There has been too much drift and delay in the way the Home Office has handled this case since January. The home secretary needs to explain urgently to parliament what she is doing to get Abu Qatada deported, and to make sure there are strong enough safeguards to protect public safety in the meantime.
"The home secretary should have acted sooner in preventing the release of Qatada in the first place. Instead of showing 'demonstrable progress' in deporting him after the European judgment, while Qatada was still behind bars, the home secretary waited seven weeks till she visited Jordan and has not appealed. Indeed, the judge who bailed Qatada noted that 'the negotiations are only at the earliest of stages'.
"Theresa May now needs to be clear and transparent with the British public. If the Jordanians have provided the assurances required to continue with Qatada's deportation, as the chair of the home affairs select committee has said he was told, what problem exists that the home secretary hasn't told us that prevent him being back behind bars?
"The home secretary needs to get an urgent grip of this case. She needs to explain what assurances she still believes are required before she can take action, why she hasn't successfully progressed Qatada's deportation and why she failed to act sooner to possibly prevent his release in the first place."
