Lib Dem peer Lord Sharkey wants a Parliamentary pardon for Turing - as the government has resisted calls to grant him a full posthumous pardon.
He said Turing and other gay men were treated with "terrible cruelty".
The government said it would not stand in the way of the bill.
Turing was convicted of gross indecency in 1952, when homosexual acts were illegal in the UK. Two years later he died from cyanide poisoning - his inquest ruled a verdict of suicide.
He was part of the team at Bletchley Park that cracked the Nazi Enigma code, vital to the allied war effort and is now widely recognised as a computing pioneer.