Benkovic had been tipped to cover the injured Caglar Soyuncu.
Rodgers says sending the 23-year-old on loan to Championship side Cardiff was still the best option.
“He's not ready to play in the Premier League, it's the brutal honesty of it," Rodgers said.
"But he needs to play games and that is his mindset as well. It was always going to be best for him and us to get out and play."
He added: “You want all your players to go away and keep fit, and you're disappointed when they come back injured. It was unfortunate for Cags. He slipped the day before he was due to play.
"Turkey told us that he'd be back in a week, but he had further scans here that showed he'd be out for longer. He's going to London to see if he wants to treat it conservatively, or have the operation."