The 23-year-old is yet to appear in the Premier League for the Tigers as he picked up the problem during a pre-season friendly against Grimsby.
Odubajo has revealed that he is hopeful of working his way into Mike Phelan's team in around six weeks' time.
"It's a huge boost to be going back outside to work every morning. I feel like a footballer again, even though I'm not back to the playing stage yet," the right-back told the club's official website.
"We have a fantastic medical team here and it feels like a blessing to be back to the stage I'm at now. I'm progressing really well and I think I've got about another month to go.
"Hopefully by the next international break I'll be in and around things, which is good. I went to see the surgeon last week and he gave the all-clear to get back outside with no real restrictions. It's a case now of trying to get back to where I was before in terms of fitness and taking it from there."
Odubajo joined Hull from Brentford in 2015.