The 26-year-old has successfully recovered from a knee injury, and returned to training ahead of Saturday’s win away to Manchester City.
But, despite taking a place on Klopp’s substitutes’ bench at the Etihad Stadium, Sturridge did not feature.
Questioned as to the status of his star striker, Klopp suggested that Sturridge is “closer” to making his first appearance since he took over at the beginning of October.
“Daniel is coming closer and closer, he has trained seven or eight times in a row,” he said.
“I will help him, and in the end it will help us.”
Klopp was keen to emphasise a patient approach to Sturridge’s rehabilitation, however, and would not give away whether the striker would play against Bordeaux on Thursday.
“He’s closer,” he repeated.
“Each session is very important, my decision you will see tomorrow, but it’s closer.
“The most important thing is he can train with no problems.
“If we have the feeling that it’s a good moment for [Sturridge to play] a few minutes or more, we will see in the game!”
Sturridge adds to Klopp’s attacking options at Anfield, with Roberto Firmino playing a starring role in the win away to City and Christian Benteke coming on as a second-half substitute.
The 48-year-old admitted that this left his side “in a good position” when it came to selection.
“Benteke is very important, when he came here he was injured, but now after many sessions with the team,” he continued.
“We are in a good position at this moment.”
However another goalscorer at the Etihad, Martin Skrtel, is unlikely to feature against Bordeaux, with the centre-back suffering from illness.
“When you have footballers, you solve one problem and you have another one,” Klopp lamented.
The Liverpool manager gave an update on the progress of Mamadou Sakho, who is currently on the sidelines with a knee injury, saying the Frenchman is “in a good way, but not ready for tomorrow.”
This could leave Klopp with Kolo Toure and Dejan Lovren as his only fit senior centre-backs.
Academy talent Daniel Cleary could be called up to his squad on Thursday.