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.
Meanwhile, Suarez has been offered a short-term sanctuary in Kosovan football while he serves his ban.
Hajvalia, who won last season's Kosovar Cup and finished sixth in the league, have made an opportunistic offer to take Suarez on loan.
With Kosovo not yet recognised as a full FIFA member, the club believe Suarez's ban would not extend to the country's league - although financial considerations would still need to be overcome for the ambitious deal to become a reality.
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 (euros, £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."
Meanwhile, Barcelona claim they have the financial power to lure Luis Suarez to the Nou Camp this summer.
Despite being sent home from the World Cup in disgrace, Suarez has been touted as the club's prime transfer target this summer and could cost up to £85 million.
Barca have already spent 42million euro (£33.6 million) on capturing 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.
The club's economic vice president Javier Faus insists there is plenty more money to buy whoever coach Luis Enrique wants.
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."
Barcelona are reported to be in negotiations with Liverpool for Suarez, with Chilean forward Alexis Sanchez as a possible makeweight in the transfer.
If the transfer is completed, Barca would be able to field a potentially terrifying front three of the Uruguayan, Lionel Messi and Neymar.