The US-based player, who rose to No 14 in the world rankings on Monday after reaching the final of the ATP 500 tournament at Queen’s Club at the weekend, last represented the country in the global team competition in September 2011.
The national squad will face Ireland in a Euro-Africa Group II relegation playoff in Centurion, starting on July 17.
“We have until 10 days before the tie to name our team to play Ireland, according to Davis Cup rules,” TSA president and acting chief executive Gavin Crookes said in a statement.
“We are doing all we can to secure a sponsor for Kevin to secure him as part of the team.”
The federation confirmed it was unable to cover the costs required by the big-serving player.
Anderson has played five Davis Cup ties since making his debut in a Group II encounter against Finland in April 2008, winning nine of the 10 matches he has played.
Needing to make himself available for the Davis Cup competition this year and next, in order to qualify for the 2016 Rio Olympics, TSA said it would accept his most recent proposal.
This could pave the way to the Games for the nation’s top tennis player, regardless of whether he turned out against Ireland.
“Anderson has voiced his commitment to his country of birth, South Africa, and TSA have said they will support Anderson and agree to sanction Anderson to play in next year’s Olympics,” the Federation said.