Politicians and campaigners insisted the technology - which allows faces captured via CCTV to be checked against watch lists in real time - is inaccurate and intrusive, adding it also infringes on an individual's privacy rights.
A letter from group Big Brother Watch has been signed by more than 18 politicians - including the likes of David Davis, Diane Abbott and Caroline Lucas - while academics, barristers and 25 campaign groups have also signed.
The organisation's director Silkie Carlo told the BBC Two's Victoria Derbyshire program: "What we're doing is putting this to government to say: 'Please can we open this debate and have this conversation.
"'But for goodness sake, while it is going on, there is now a surveillance crisis on our hands that needs to be stopped urgently.' "
However, the companies behind the technology insist it protects the public with the ability to help catch terror suspects.
The Home Office added it supports the police ""as they trial new technologies to protect the public, including facial recognition, which helps them identify and locate suspects and criminals".