The Wales defender was injured during Leeds' Premier League draw with Brentford and is now facing a lengthy spell on the sidelines.
Farke has admitted the news was a major setback for Leeds, with Rodon having started both of the club's league matches so far this season.
"Yes, bad news," Farke said ahead of Saturday's away match against Brighton.
"Joe has a hamstring injury. He will be out for, at the moment, an assessment of eight to 10 weeks, so he will definitely not play before the next international break.
"Also, probably not the first games after the international break. A blow for us because he is a key player for us.
"But it is football and sometimes you have to deal also with injuries. He will definitely miss the next eight to 10 weeks."
Rodon's absence will be a significant problem for Farke as Leeds prepare for a demanding period in the Premier League.
The injury is also likely to rule the 28-year-old out of Wales' upcoming Nations League fixtures against Portugal and Denmark later this month, as well as the matches against Norway and Denmark in October.
Leeds will now have to find a way to cope without one of their regular starters when they travel to the American Express Stadium to face Brighton on Saturday.
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