England's leading bowler in tests with 685 wickets, Anderson suffered the injury playing for Lancashire last week. He is unlikely to play in the one-off test against Ireland before the Ashes after the uncapped Josh Tongue was added to the squad.
"Yes, definitely," Anderson told the Telegraph newspaper on Wednesday when asked if he would be fit for the opener against Australia at Edgbaston on June 16.
"I have no issues running and bowling again and just keeping my fingers crossed I can carry on going as I have been and will be ready for the Ashes.
"The idea is to try and step it up here at Lancashire over the next couple of days and then meet up with England on Sunday and see where I get to. I am in the squad for the Ireland test but we have to make a call on that nearer to the time."
Anderson's availability is crucial for England, who are already without injured fast bowler Jofra Archer.
England lost 4-0 in the last Ashes series in Australia.
