Cole, who was on loan at French outfit Lille last season, suffered a hamstring problem on the opening day of the season in the clash with West Brom, and has been missing ever since.
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The Reds boss indicated that the winger's rehabilitation is progressing well, but believes the 30-year-old is still a "little way" off a return to first-team action.
The Northern Irishman told the club's official website: "Joe is working well and the medical department are pleased with his progress.
"He was out onto the pitch over the weekend doing some work, so there's still a little way to go with Joe yet but he's on the right track, which is important."
Assaidi is a new signing at Anfield after joining from Dutch side Heerenveen in the summer for a fee reported to be around £2.4 million, but is yet to make an appearance due to injury.
The Moroccan winger was forced to withdraw from Monday's reserve match with Manchester City, which ended as a 2-2 draw, after he felt tightness in his groin.
The ex-Swansea manager is confident that the 24-year-old, who scored 20 goals in 79 games for his former club, will take a part in training with the rest of squad in the near future.
The 39-year-old added: "He should train today. He was due to play in the reserve game, but after training with the reserves in preparation for the game on Sunday, he felt a little bit of tightness.
"We're hoping in the next couple of days he'll be ready to train and re-join the group."