The 26-year-old has not competed since pulling his right hamstring at the Diamond League meet in Paris on 6 July.
But he said he still hoped to be fit for next month’s World Championships in Moscow, and has been given more time – until 30 July – to recover.
Rutherford said: “I’m working as hard as I can.”
The Englishman was not named in the GB team announced this week for the Worlds, which start on 10 August.
British Athletics performance director Neil Black, who has said the Olympic champion does not have to compete prior to Moscow in order to prove his fitness, will pick between Rutherford and Chris Tomlinson.
The Anniversary Games is a three-day meeting being held at London’s Olympic Stadium between 26-28 July. It marks a year since the opening ceremony of the 2012 Olympic Games.
Rutherford had been hoping to compete alongside British team-mates Mo Farah and Jessica Ennis-Hill, who is also struggling with an ankle injury, on the second day.
The trio all claimed gold medals within an hour of each other on 4 August, which became known as ‘Super Saturday’, in one of the most memorable moments of the Games.