Mr Floyd was killed by police officers in Minnesota in 2020 during an unrelated arrest, which sparked months of protests about racial justice and police brutality.
Allison Mathis, who defended Mr Floyd in the 2004 case, received a letter from the Texas Board of Pardons and Paroles, which stated that it had “completed their consideration of your client’s application requesting a Full Pardon and have voted to recommend clemency”.
Mr Floyd was arrested on 5 February 2004, by then-Houston undercover narcotics detective Gerald Goines, who claimed Mr Floyd had given a second suspect 0.03 grams of crack cocaine to sell.
The man to whom Mr Floyd allegedly gave the drugs was a police informant who sold the drugs to the detective as part of a sting operation and was never arrested or identified, according to the petition written by Ms Mathis.
Mr Floyd pleaded guilty to a drug charge and was sentenced to 10 months in state jail.
Mr Goines has been charged withthe murder of Rhogena Nicholas and Dennis Tuttle during a botched drugs raid on a home in Houston.
A police colleague of Mr Goines, Steven Bryant, has been charged with falsifying records to help him allegedly cover up an illegally obtained “no-knock” warrant for the raid.
“We lament the loss of former Houstonian George Floyd and hope that his family finds comfort in Monday’s decision by the Texas State Board of Pardons and Paroles to recommend clemency for a 2004 conviction,” Harris County District Attorney Kim Ogg Ogg said in a statement.
Ben Crump, the Floyd family lawyer, said he wants Mr Abbott to grant Mr Floyd a full pardon.
“This drug charge, which led to George Floyd’s conviction based on false evidence, helped to unravel his life,” Mr Crump said in a statement.
“Similarly, tens of thousands of Black lives are ruined by a criminal justice system that uses the war on drugs to target Black people, force them into felony pleas, incarcerate them, take away their voting rights, and destroy their families.”
Mr Floyd died on 25 May 2020 following his altercation with police in Minneapolis.
Former police officer Derek Chauvin, who pressed his knee into Mr Floyd’s neck for more than nine minutes during the arrest, was convicted in April of second-degree unintentional murder, third-degree murder and second-degree manslaughter.
He will serve more than 22 years in prison.
Three other police officers involved in the fatal arrest have been charged with aiding and abetting second-degree murder and aiding and abetting second-degree manslaughter, and face trial next year.