The Alan Turing law could see thousands of men absolved for crimes they would be innocent of today.
Proposals to introduce new legislation which would pardon gay men convicted under historical gross indecency laws will be brought forward “in due course”, the government has said.
A proposal was included in the 2015 Conservative manifesto and was supported by the Labour Party.
The Alan Turing law – named after the World War II code-breaker – will pardon all men convicted for being gay.
Turing received posthumous royal pardon back in 2013, 61 years after he was charged at Manchester police station for “gross indecency”.
The pioneering mathematician was chemically castrated by a series of injections of female hormones and he took his own life two years later, in 1954.
Mr Turing’s great-niece Rachel Barnes told The Independent she had been “disappointed that there had been no progress” since she and others had launched a petition last year to get her great-uncle’s pardon extended to all men convicted under the historic laws.
She said: “I will give my backing towards any progress.
“It’s something the family of Alan Turing have always, always backed and we really do want justice for everybody who was affected by the anti-gay laws.
“I am very pleased to hear the current administration will give it their backing.”
Campaigners and Mr Turing’s family delivered the petition to Downing Street before last year’s general election.
Homosexuality was decriminalized in England in 1967.