Atapattu is embroiled in a stand-off with Sri Lanka Cricket officials following his criticism of national selectors during the first Test, in Brisbane.
After his half-century in the first innings at the Gabba, Atapattu said in a press conference that Sri Lankan selectors were "muppets headed by a joker," referring to panel chairman Ashanta de Mel.
The Sydney Morning Herald newspaper said Atapattu spoke to officials of Sydney's St George club between the first and second Tests regarding a possible move to Australia.
The newspaper said the former Sri Lankan captain could be in Sydney before Christmas and be available to play for St George, in Sydney's southern suburbs.
Atapattu is expected to face a disciplinary hearing over the outburst in Sri Lanka on his return home from the Australian tour.
But Sri Lankan team sources told the newspaper the 36-year-old opening batsman is considering announcing his retirement from international cricket before the country's forthcoming Test series against England.
The Herald said Atapattu is also understood to have held preliminary discussions with an Australian media organisation regarding commentary work during the triangular one-day series between Australia, Sri Lanka and India in February.
The right-hander was initially overlooked by Sri Lankan selectors for the Australian touring party, only to be called into the squad after the intervention of the Sri Lankan sports minister.
Prior to Brisbane, Atapattu had not played a Test match since December 2005.
But he went on to score 51 in the first innings to keep his place in the team.
In 89 Tests Atapattu has scored 5,397 runs at 38.83, with 16 hundreds.