The Swans defender was heavily critical of Suarez in an extract from his Premier League diary recounting last season, their first in the Premier League, published this week. Williams was quoted as saying the Uruguay international was
"streets ahead of any player I've truly disliked" and he wanted to "knock him out" for the player's diving antics.
Rodgers was taken aback when he read those comments and has spoken to Williams, who in turn has talked to Suarez, in an attempt to prevent the situation escalating ahead of Sunday's match at the Liberty Stadium.
"I have spoken to Ash and Ash has spoken with Luis so there is no issue there," said Rodgers, who along with Joe Allen returns to the club they left in June.
"When I saw it I was surprised because in my time there because Ash, knowing his character, is a real good man with good intelligence and maturity and a good footballer player.
"When I saw those comments I spoke to him and got the full explanation of how it has seemingly got out there and he was very quick to want to apologise for that.
"That is done. There is no problem there and I am sure they will shake hands on Sunday and get on with it."
The fact Rodgers was able to use his good relationship with his former club to intervene in the matter has no doubt helped avoid any lingering ill-feeling. Last season Suarez's obvious disdain for Patrice Evra, whom he was found guilty of racially abusing and banned for eight matches, manifested itself in the refusal of a handshake with the Frenchman at Old Trafford in February. It was one of a number of high-profile public relations disasters by the club in respect to the affair and Rodgers was keen to avoid any chance of things getting out of hand at the Liberty Stadium. Rodgers has already lost to his former club, at Anfield a month ago, when they ended Liverpool's defence of the Capital One Cup. Visiting fans cheered former midfielder Allen but chanted
"We don't need you any more" at Rodgers.
The Liverpool boss does not expect to encounter much hostility but accepts, with points at stake, there may not be a warm welcome.
"It is a real special place Swansea. I spent two fantastic years there and had a wonderful rapport with the supporters," he added.
"The club was brilliant for me and hopefully I gave them something back while I was there.
"I never said I was going to be there for many years but it took a truly unbelievable club like Liverpool to draw me away.
"Whatever reception I get is irrelevant, the important thing is I know the good people there and I am sure time will heal any bad thoughts.
"But we are going there to keep our (seven-match) unbeaten in the league going."
Key to Rodgers' success in south Wales was his close friendship with chairman Huw Jenkins. A move to a club Liverpool's size makes those kind of relationships more difficult but Rodgers feels it is working well with United States-based principal owner John Henry and chairman Tom Werner.
"When I met Huw I was very impressed. It was a hand-in-glove fit me going there," said the Reds boss.
"It was so refreshing speaking to a chairman who had a clear identity of how the club worked all based around simplicity and a philosophy that has been going for many years.
"I believe it can be replicated at Liverpool. Simplicity is the key.
"That is what this club was based on over many years: good players, good coaching, good people and good values and ethics.
"As long as we can continue to keep it that way and not complicate things we will be fine.
"This is a different kettle of fish altogether, a world-renowned football club where everyone has an opinion but as long as we manage that and know where we are going we will be fine.
"I don't speak to the owners every day but we have communication every other day by phone or email.
"You have to understand there is a board who have brought you in to do a job and I respect that.
"I am here to try to keep it moving in the same direction and in order to do that you need to communicate well."