The Wales international was forced off after just 34 minutes of Sunday's 1-1 Premier League draw with Brentford, having started the season well with appearances in the opening league win at Nottingham Forest and a Carabao Cup victory over the same opposition.
"Bad news, serious hamstring injury," Farke said. "He will be out for eight to 10 weeks. Definitely not playing before the next international break."
The 28-year-old's absence leaves Leeds short at centre-back ahead of Saturday's trip to Brighton (15:00 BST), with Farke suggesting James Justin could be asked to fill in there instead of his usual role covering at left-back.
"A big blow for us because he's a key player," Farke said. "Puts more pressure on us to get a wing-back — we may need James Justin at centre-back.
"It is important Joe comes back as quick as possible. Need a bit of luck, but we also have the January window to react if necessary."
Justin has been deputising at left-back following a pre-season hamstring injury to Sweden international Gabriel Gudmundsson, a gap Leeds moved to address on deadline day by signing French defender Melvin Bard. Farke gave his first assessment of the new arrival, along with fellow deadline-day addition Jean-Mattéo Bahoya, ahead of the Brighton fixture.
"He has played many, many games in Ligue 1," Farke said of Bard. "Also like that left-back or left wing-back cover that we needed. Of course, it's early doors because he's had just more or less two training sessions together with us, but very experienced player but also a good age."
On the 21-year-old Bahoya, who arrives from Eintracht Frankfurt having recently played in the Bundesliga, Farke added:
"Obviously a young prospect anyhow, recently played in Germany. Very great character, very quick player, so lots of potential. He can take players on, but of course, he's very, very young in age so he's different in comparison to Melvin. I'm pleased that both of them are with us."
Leeds United travel to Falmer Stadium looking to extend their unbeaten start to the 2026/27 campaign, having gone through their opening matches without defeat under Farke.
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