Sturridge has missed Liverpool's last five games due to a thigh injury sustained during the previous international break at the start of last month.
The 25-year-old will again be sidelined for his club's UEFA Champions League clash with Basel on Wednesday, but could make a return to Rodgers' squad for their Premier League meeting with West Brom at the weekend.
Rodgers was openly critical of England's handling of Sturridge after he suffered the injury in training, insisting it "could have been avoided".
And the Northern Irishman has now urged Hodgson to refrain from picking Sturridge in his squad for England's qualifying double-header with San Marino and Estonia, which will be announced on Thursday.
"Daniel won't be fit [for England]," Rodgers told his pre-match media briefing in Switzerland.
"Whether he is available for the weekend or not, he won't be fit. We are hoping he can develop over the next 24 hours and train with the team on Thursday and be available for some involvement and be in the squad on Saturday.
"But if you are talking about players' availability, they need to be fit to play and Daniel will not be fit to play."
Rodgers went on to say he has not spoken with Hodgson since Sturridge picked up his injury.
"We have good dialogue and he knows he can call and speak about any of the players," he added.
"We will see what happens. My focus is on protecting the Liverpool players and at this moment in time my focus is on getting him fit for Liverpool."
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