He has not played since August due to a groin problem suffered in an England training session, which prompted an angry Rodgers to say the injury "could have been prevented".
Sturridge did not travel to Switzerland for Wednesday's Champions League Group B game at Basel and is doubtful for Saturday's Premier League game at home to West Bromwich Albion.
However, Rodgers was quoted by British media on Tuesday as saying: "Whether he is available for West Brom or not, he won't be fit. We are hoping he can train with the team on Thursday.”
"We hope he'll be available on Saturday. But if you are talking about a player's availability for England they need to be fit to play and Daniel will not be fit to play,” he added.
"It's something we have stressed. My focus is on protecting Liverpool players and, at this moment, my focus is on getting him fit for Liverpool,” he said.
England host San Marino at Wembley next Thursday and travel to Estonia the following Sunday.
Liverpool French defender Mamadou Sakho is also set for a spell on the sidelines after suffering a thigh injury.
"His injury came in a training session. It's unfortunate -- he's just felt a pain in his thigh," Rodgers said on the club website (www.liverpoolfc.com).
"He went and had a scan on it which showed he's got a slight tear in there, which will be probably two to three weeks,” he added.
The centre-back also faces possible disciplinary action from the Merseyside club after he walked out of Anfield before the derby against Everton when he found out he was not in the match-day squad. He later apologised on Twitter.

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