Theresa May, the home secretary, will conduct a "proper, sober, thoughtful review" into Britain's extradition treaty with the US amid anger at a series of deportations, David Cameron has announced.

The prime minister told MPs the home secretary would take account of the views of parliamentarians after he was asked about the case of Christopher Tappin.

The retired company director from Kent is due to be flown to the US on Friday to face allegations of selling arms to Iran. Tappin, 65, admits shipping batteries that can be used in Hawk air defence missiles but says he thought they were for use in the car industry. He said he had no idea about their eventual destination.

Tappin's case was raised by his MP, Jo Johnson, at prime minister's questions on Wednesday. Johnson, the MP for Beckenham and brother of the London mayor, Boris Johnson, said: "US marshals will on Friday escort my 65-year-old constituent Chris Tappin from Heathrow to a jail in Texas, where he will face pressure to plea-bargain in order to avoid lengthy incarceration pending a financially ruinous trial for a crime he insists he did not commit.

"Could the prime minister say what steps he is considering to reform the US/UK extradition treaty that been so unfair to the likes of Gary McKinnon and now my constituent, Mr Tappin?"

The prime minister indicated the government would not block the extradition of Tappin. He said: "I quite understand why [he] raises this case of his constituent. In the case of Chris Tappin obviously he has been through a number of processes including the magistrates court and the high court. The home secretary has thoroughly considered his case."

Cameron cited an independent report last year by Lord Justice Scott Baker, which said that the 2003 Extradition Act was not "lopsided" or biased against British citizens. Critics have said that the act, drawn up in haste after the 9/11 attacks, is unfair because British citizens do not enjoy the same level of legal protection as US citizens.

Gary McKinnon, the alleged computer hacker who has Asperger's syndrome, faces extradition under the treaty.

The prime minister said: "[Jo Johnson] raises the point more generally of Sir Scott Baker's report into the extradition arrangements, which he has made and we are now considering. He did not call for fundamental reform."

But Cameron said May would lead a thorough review of the extradition treaty. "The home secretary is going to carefully examine his findings and also take into account the views of parliament that have been expressed in recent debates.

"Of course balancing these arrangements is absolutely vital. But I think it is important that at the same time we remember why we enter into these extradition treaties, which is to show respect to each other's judicial processes and make sure that people who are accused of crimes can be tried for those crimes and Britain can benefit from that as well. So a proper, sober thoughtful review needs to take place and this case shows why."