Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
The Reds boss stressed an apology would be required in order for him to return to the fold, after Suarez claimed Rodgers had reneged on a deal to allow him to leave for a Champions League side.
It is understood Suarez is now back in training with his senior team-mates, but the club would not confirm how - or even whether - an apology had been made.
Last week the Uruguay international was told he was to work away from the main group after Rodgers said his attitude and performances had not been up to the required standard during his protracted transfer dispute with the club.
Arsenal have had two bids for the player rejected, including one of £40,000,001 in an attempt to activate a clause in the South American's contract.
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.
But since Suarez arrived back at the club's Melwood training base there appears to have been some kind of reconciliation, but Rodgers admitted on Thursday 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."