The 23-year-old Dane hasn't played since he was hurt in the semi-finals at Stockholm last year.
There had been indications that he was nearing a return when photos surfaced on social media this week of him playing practice sets in Spain with Carlos Alcaraz.
Seven-time Grand Slam champion Alcaraz has been sidelined himself since April with a wrist injury.
But Alcaraz posted on social media on Thursday that he was back - just in time for his title defense at the final Grand Slam of the year which starts in New York on August 30.
US Open organizers said Rune's place in the main draw would be taken by Argentina's Facundo Diaz Acosta.