The 27-year-old Uruguayan was also expelled from the World Cup and handed a nine-match competitive international ban for the incident in a Group D match.
The Republic of Abkhazia is located within Georgia, but claimed independence following Georgia's brief war with Russia in August 2008.
The territory has not been widely recognised by the international community while its local football federation is not part of FIFA.
Gennady Tsvinariya, the president of the club, said Suarez will be able to take part in the Abkhazian national championship.
"Of course we can not offer the Uruguayan footballer a financial package that he is used to, but the Abkhazian championship is at its peak," Tsvinariya told the ITAR-TASS news agency.
"We will be able to offer Suarez the chance to maintain his match fitness."
Earlier, Hajvalia from Kosovo offered the Liverpool striker, who is reportedly in talks with Spanish giants Barcelona for a possible transfer, the chance to play for their side while he serves his ban.
The Kosovo Football Federation is also not a member of FIFA and are therefore not governed by the world football body.
(Editing by Sudipto Ganguly)
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