Sturridge went under the knife in New York to fix an issue believed to be behind the various muscle problems he suffered but looked to be moving well when he was put through his paces in a solo training session at Melwood on Thursday.
"There's no timeline on Daniel. He has come back and it's just about him working hard now to get somewhere near the level of the group. There's no time on it," manager Brendan Rodgers told the club's website (www.liverpoolfc.com).
"He came back and looks well -- he's really excited and happy to be back among his team mates.
"We'll just have to take that week on week, rather than put any sort of pressure on a time for him," the 42-year-old added.
Meanwhile, midfielder Lazar Markovic appears set to join Turkish side Fenerbahce on a season-long loan, according to British media reports, with the clubs said to have agreed terms for the 21-year-old.
The Serbia international joined Liverpool from Benfica for a reported 20 million pounds ($31 million) last season but made just 11 league starts as he struggled to cope with the physical nature of the Premier League.
He has not featured in the squad in Liverpool's three matches this season.
Markovic will become the second player to leave Anfield on loan this week should the deal go through, following Italian striker Mario Balotelli's departure for AC Milan on Thursday.
Unlike Balotelli, however, Markovic could still feature in Rodgers' long-term plans as the loan agreement does not contain a clause to make the move a permanent one, the reports added.
(Reporting by Simon Jennings and Shravanth Vijayakumar in Bengaluru; Editing by John O'Brien)
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