The Uruguay 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.
Suarez will definitely not feature in Saturday's Premier League opener at home to Stoke. He has six games left to serve of the 10-game ban he received for biting Branislav Ivanovic of Chelsea last April.
Despite his oft-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) Brendan 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 Suarez is ready to fight for his first-team place."
After last weekend's friendly defeat to Celtic in Dublin, Rodgers confirmed Suarez would need to say sorry before any reintegration took place.
"Initially there will be a recognition that (there needs to be) an apology to his team-mates and the club," he said.
"When he is back from his international trip (Suarez went to Japan with Uruguay in midweek) we will assess it from there.
"He is a world-class striker and when he is committed to the cause we will welcome him back with open arms."
Initially there was some suggestion while he was on international duty that the player's stance had mellowed, with Uruguayan newspaper El Observador quoting him as saying: "For now, owing to all the affection of the people, I would be staying."
However, after the 4-2 victory over Japan, in which Suarez scored, the Japanese media reported Suarez as claiming he had never uttered those words.
Since Suarez arrived back at the club's Melwood training base, there appears to have been some kind of reconciliation, but Rodgers admitted he did not want give more publicity to a saga which has rumbled on all summer.
"There has been so much written and said about Luis over the last few months I don't really want to answer any more questions on Luis Suarez," he said on Thursday.
"What I want to concentrate on is the players that are available for this weekend (against Stoke) so I won't be speaking any more about Luis Suarez until he is available.
"There is nothing to be said that hasn't already been said. I am sure people are fed up of listening to it and reading about it."