And with Glen Johnson and Dan Agger set to return, Brendan Rodgers could soon have a near fully-fit Reds squad to choose from.
Lucas was not expected to return before March after injuring his knee against Aston Villa in January - but his progress is 'weeks' ahead of schedule.
Reds head of performance Glen Driscoll revealed: "Lucas Leiva was due to be in a brace for six weeks, according to specialists' opinion, but yesterday he was outside running just over three weeks in, which is fantastic and shows just how ahead of schedule he is."
And the news was equally as positive for Sakho and Enrique.
"Mamadou Sakho actually had a tear in his tendon and not the muscle of the hamstring as initially reported. For a whole host of reasons tendons take longer to heal, but he is working hard and progressing," Driscoll said.
"Jose Enrique also had a complex knee issue and following surgery is in the process of achieving his final strength gains before he can return to training.
"Without providing specific dates, I would hope that by the end of the month we should start to get the likes of Jose, Lucas and Sakho back."
Agger and Johnson are both in contention to play at Fulham on Wednesday evening.
"Glen and Daniel have come back into training, which is brilliant for us," Brendan Rodgers told the Liverpool Echo.
"They certainly look very strong in training.
"Whether the midweek game is going to be too soon for them, we will see. But it's great to have them back and see their qualities."