Sahin has been strongly linked with a loan move to either Arsenal or Liverpool throughout the summer, but the Turkish international appears to have chosen the latter as his preferred destination.
Rodgers expects the deal to be done within 24 hours, and the Northern Irishman was quick to praise his prospective signing.
"I believe, from the club, everything has been organised and sorted in terms of paperwork so we're hoping in the next 24 hours that can be done," he said.
"He is a great technician. He was player of the season in Germany, captain of Dortmund and fantastic at a very young age. He got a move to Real Madrid and obviously through injury it has not quite worked out for him.
"But he is a wonderful player technically and tactically who understands the game and we hope his extra quality coming in will help the group."
Madrid manager Jose Mourinho has already backed Sahin to shine in the Premier League.