The Uruguayan international had been frozen out of first-team sessions after publicly demanding a transfer, but the rift between player and club now appears be closing.
However, he will definitely not feature for Saturday's Premier League opener at home to Stoke because he has six games left to serve of the 10-game ban he received for biting Branislav Ivanovic of Chelsea last April.
Despite Suarez's regularly-stated desire to play Champions League football and a pair of bids from Arsenal, Liverpool have remained firm in their stance that he is not leaving and it appears increasingly likely that he will still be at Anfield when his suspension is completed.
Sky Sports reporter Vinny O'Connor said: "Luis Suarez has apologised to (manager) Brandan Rodgers and apologised to his team mates over his general conduct.
"He has started his re-integration into the Liverpool first-team and trained with them on Friday.
"Over a week ago, Brendan Rodgers said he would need an apology from Luis Suarez, as would the fans and his team mates, after the apparent attempt by the player to force a move away from Anfield. It appears that the apology has come.
"Good news, in many ways, for Liverpool football club. It looks like things are beginning to settle down and Luis Suarez is ready to fight for his first-team place."