Babar Ahmad is challenging extradition to the United States where he could be jailed for life.
His family is urging the Director of Public Prosecutions to try him in the UK where the alleged offences occurred.
Mr Ahmad is waiting for a final ruling on his case from the European Court of Human Rights.
The US alleges that Mr Ahmad was responsible for a global fund-raising campaign for extremists in Afghanistan and Chechnya through a website operated from south London but technically based in the US. He is also accused of having obtained information about US Navy ships and their movements in the Gulf. He denies the allegations.
Mr Ahmad was first arrested by the Metropolitan Police in 2003 but then released without charge. The force later paid him £60,000 compensation after admitting that he had been assaulted during arrest.
In 2004 he was arrested again to face extradition to the US and since then he has been held in maximum security prison units. Two British judges who have dealt with Mr Ahmad’s case have raised concerns, with one describing his continuing detention as an “ordeal”.
The e-petition to put Babar Ahmad on trial in the UK topped 100,000 after a community campaign run by his family attracted the support of celebrities including boxer Amir Khan and comedian Mark Thomas. As of Thursday morning, the petition was the fourth most popular on the e-petition website.
The petition says: “In June 2011, the… Joint Committee on Human Rights urged the UK government to change the law so that Babar Ahmad’s perpetual threat of extradition is ended without further delay.
“Since all of the allegations against Babar Ahmad are said to have taken place in the UK, we call upon the British Government to put him on trial in the UK and support British Justice for British Citizens.”
In a statement, Mr Ahmad’s family said: “Our solicitors have prepared a fresh file of evidence to the Director of Public Prosecutions Keir Starmer QC proving that the CPS should have prosecuted Babar in 2004.
“Over 100,000 members of the British public now ask the DPP to confirm that he will instruct independent counsel to conduct an urgent full re-review of this case so that Babar Ahmad can be put on trial in the UK.
“We also ask the coalition government to confirm that a full Parliamentary debate will be held about Babar’s case with a view to putting him on trial in the UK.”
“We are overwhelmed by the amount of support for Babar’s case. This shows that the British public is still prepared to stand up against injustice.”
Under the e-petition rules, if an issue tops 100,000 signatures before a deadline, it becomes eligible for a possible debate in the House of Commons. Last month saw the first such debate when MPs discussed releasing documents relating to the Hillsborough disaster.
Although the e-petition has reached the target, it is far from certain that MPs will pick up the issue. E-petition debates can only take during parlimentary time allocated for backbench motions – and most MPs have their own topics which they wish to have debated.
The rules also state: “If the subject of the e-petition is currently going through legal proceedings, it may be inappropriate for a debate to be held.”
The current most popular e-petition, to withdraw benefits from anyone convicted of involvement in London’s riots, has not been debated by MPs.