The ex-Leyton Orient and Brentford defender suffered the injury in a pre-season friendly against Grimsby Town.
Meanwhile, the club have confirmed they have taken former Stoke City forward Peter Odemwingie, 35, on trial.
The Nigeria international, who could feature for Steve Bruce’s side in a friendly with Mansfield on Tuesday, is a free agent after being released by the Potters at the end of last season.
Odemwingie ended the campaign on loan at Championship side Bristol City after recovering from a long-term knee injury.
Odubajo’s injury has reduced Hull to just 14 fit senior outfield players with less than four weeks until they host Premier League champions Leicester City on the opening day of the season.