Striker Sturridge played 89 minutes of England's 1-0 friendly win over Norway at Wembley on last Wednesday before returning to training 36 hours later and picking up the injury which could keep him out for three weeks.
The 25-year-old missed England's 2-0 Euro 2016 qualifying victory over Switzerland on Monday.
"We're obviously disappointed because we feel it was an injury which could have been prevented," Liverpool boss Rodgers said.
"He has worked so hard over pre-season and looked very fit and strong in our last game against Tottenham. It could be up to three weeks, which is obviously disappointing," Rodgers, whose side play Aston Villa on Saturday, said at a news conference later.
Sturridge, who has scored 32 goals in 46 Premier League appearances for Liverpool, is used to having two days to recover after matches and Rodgers believes international teams should take players' needs into more consideration.
"I think clubs work differently at times to international teams. It's more the recovery strategy," the Northern Irishman said.

"When we look at our players here we look at them individually in terms of what their needs are,” he added.
"Fast players would have a second day recovery while other players can work on that day. When you are that type of player like Daniel Sturridge, Raheem Sterling and boys like Danny Welbeck you need to recover them," he further said.
Sturridge was not the only Liverpool casualty after the international break. Emre Can and Joe Allen picked up knocks while playing for Germany and Wales respectively.
"We've got Emre Can back from Germany, who could be out for up to six weeks because of his ankle," Rodgers said.
"We've got Joe Allen back from playing on (an) awful pitch (in Andorra) and he has an issue with his knee. We just need to assess that for the weekend. Those are the types of things that can happen in games and it's unfortunate,” he added.
Manchester United defender Phil Jones is also waiting to find out the extent of the hamstring injury he suffered during England's win in Basel after playing on for 15 minutes with the problem.
The loss of defender Jones will come as a blow to Louis van Gaal when United host Queens Park Rangers at Old Trafford on Sunday hoping to pick up their first win of the season.

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