The 33-year-old bowler picked up the injury playing in the Indian Premier League and returned home after completing his third match on April 14.
"Scans showed the injury was not serious and he will undergo intensive rehabilitation with the aim of him being available to take part in the Australia A Series and we expect he will be fit to play in the Ashes," Cricket Australia chief medical officer Justin Paoloni said.
Harris has picked up 47 wickets for Australia in 12 Tests and is part of the squad for the first of the back-to-back Ashes series against England, starting at Trent Bridge in Nottingham on July 10.
Another fast bowler, Mitchell Starc, who underwent ankle surgery to remove bone spurs, is on schedule to be ready for the Champions Trophy in June.
"Mitchell Starc is recovering as expected from arthroscopic ankle surgery," Paoloni added.
"Currently he is meeting all clinical timeframes and with this progress the target is to play ICC Champions Trophy and then the Ashes."
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