Suarez was sent home from the World Cup in disgrace after biting Chiellini during Uruguay's 1-0 win over Italy on June 24.
The Liverpool striker, who had bitten two opponents on the field of play before, was suspended from ''all football-related activity'' for four months by FIFA as a result.
After reflecting on the incident in his Montevideo home for the last few days, Suarez decided to release a statement on Monday apologising for his latest indiscretion. It is understood Liverpool had no involvement in Suarez's apology.
The statement, entitled ''My apologies to Chiellini'', was published on Suarez's Twitter account.
It read: ''After several days of being home with my family, I have had the opportunity to regain my calm and reflect about the reality of what occurred during the Italy-Uruguay match on 24 June 2014.
''Independent from the fallout and the contradicting declarations that have surfaced during these past days, all of which have been without the intention of interfering with the good performance of my national team, the truth is that my colleague Giorgio Chiellini suffered the physical result of a bite in the collision he suffered with me.
''For this: I deeply regret what occurred. I apologise to Giorgio Chiellini and the entire football family. I vow to the public that there will never again be another incident like this.''
Within an hour of the tweet being posted, Chiellini accepted Suarez's apology.
".@luis16suarez It's all forgotten. I hope FIFA will reduce your suspension," the Juventus defender responded on Twitter.
Uruguayan president Jose Mujica has expressed his dismay at the length of the punishment, which also came with a £65,000 fine.
Mujica, who had said previously that Suarez was being punished by FIFA because of his humble background, continued his verbal assault at a reception for the Uruguay team after they returned home from the World Cup following their 2-0 defeat to Colombia in the last 16.
Asked by a journalist what his lasting memory of the World Cup would be, Mujica said: ''FIFA are a bunch of old sons of bitches.''
The president then covered his mouth to feign shock at what he had just said, but when asked by the journalist if he wanted to rectify his comments, he responded: ''Publish it.''
Mujica then conceded that Suarez deserved to be punished for his bite, but criticised the severity of the ban.
''They could have punished him, but not given him this fascist ban,'' added Mujica.
FIFA refused to respond to the president's remarks.
With Suarez's ban also affecting his future at club level, Liverpool are expected to consider offers - with Barcelona seemingly heading the queue.
The Catalan club claim they have the financial power to lure him to the Nou Camp this summer - but the same may not be true of surprise suitors KF Hajvalia.
The latter have offered Suarez a short-term sanctuary in Kosovo - not a full member of FIFA - while he serves his ban.
The club won last season's Kosovar Cup and finished sixth in the league and have made an opportunistic loan offer.
Director Xhavit Pocolli told Kosovan daily Sport Plus: ''Suarez can't play in the next four months. As we are not part of FIFA yet, I think he can play in Kosovo, so we have an offer that we will send to Liverpool.
''We offered 30,000 euros (£24,000) and a salary of 1,500 (£1,200) for each month. This might sound ridiculous to him, but that is all we can do.
''If he is willing to come and play for us, he is welcome. As we are not part of FIFA, it would be ideal for him.''
A significantly heftier fee of up to £85 million has been touted for a permanent move, but Barca vice-president Javier Faus insists that will not put the club off.
Barca have already spent 42million euros (£33.6 million) to sign midfielder Ivan Rakitic from Sevilla and goalkeepers Marc-Andre ter Stegen and Claudio Bravo from Borussia Monchengladbach and Real Sociedad respectively, but pocketed around £30million in the sale of Cesc Fabregas to Chelsea.
Speaking at the announcement of a sponsorship deal between the club and appliances company Beko, Faus explained the club's finances were in rude health.
''In the last three years Barca have been earning money and reducing their debt at a large rate and are stronger financially,'' he said.
''We can realise all the sporting challenges and signings our coaches and sporting director want.''
Chile forward Alexis Sanchez could be used as a makeweight in a transfer, which would leave Barca able to field a potentially terrifying front three of Suarez, Lionel Messi and Neymar.
However, the club are also expected to sign a central defender, with long-standing captain Carles Puyol hanging up his boots at the end of last season.