The 30-year-old has recovered from a lengthy layout on the sidelines and looks set to return to the first team after playing an hour for the Under-21s and grabbing a goal
The 30-year-old midfielder, who spent last season on loan at Ligue 1 side Lille, has been on the sidelines for six weeks after aggravating a hamstring strain during the Anfield club's opening day clash with West Brom.
Cole suffered the hamstring strain during Brendan Rodgers' first competitive game against FC Gomel in August, but worsened the injury after coming on as a substitute against the Baggies.
However, the attacking midfielder was able to complete 60 minutes of play for the Under-21s at the club's academy in Kirkby, in which he opened the scoring to beat Fulham 3-0 on Monday.
The former England international has been able to step up his recovery and could challenge for a spot in the Europa League clash against Udinese on Thursday, with Rodgers looking to rotate his squad.