Reds boss Brendan Rodgers feared the midfielder would be out for up to six weeks when he suffered a hamstring injury against West Ham earlier this month.
However, just three weeks later the 33-year-old is recovering well and it is thought he could be back early the new year, although the January 1 home clash against Hull City looks like coming too soon for him.
"Steven is working very well with the staff," Rodgers said. "He's already back running and kicking a ball. Hopefully in the next week to 10 days we'll have him back training with us fully."
Liverpool have suffered an injury blow ahead of the Boxing Day trip to Manchester City after Rodgers confirmed young full-back Jon Flanagan is likely to miss a few weeks with his own hamstring injury.
"It looks like he will be out for a number of weeks with a strain," he added. "It's a blow because Jon has been brilliant for us since he came into the side."