A woman in her 40s tested positive in Mount Gambier after purportedly travelling across the border to Casterton.

SA Health has since reduced the size of the border travel corridor with Victoria to 30km east of the border and 40km west.

This excludes Victorian towns such as Nhill, Edenhope and Casterton from travelling across to South Australia as part of the cross border community.

There have also been unconfirmed reports that the woman who tested positive in Mount Gambier had not fully disclosed her travel path, and was originally from Melbourne.

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In a press conference on Monday afternoon, South Australia police commissioner Grant Stevens was unable to confirm whether the woman originally travelled from Melbourne.

"We are certainly investigating this person's movements up to and including the time they tested positive," he said.

"It is critically important we identify every location that person has been to so we can establish exposure sites, or whether there may have been close contacts."

South Australia's chief public health officer Nicola Spurrier said South Australia's contact tracing team was working to establish any possible exposure sites as a result of the positive case.

"We just need to be very cognizant that not everyone may be in the same situation that we are in," she said.

"It is very important that we keep that as very privileged, public health information.

"We will use other ways of corroborating that information so that we can make sure that people across South Australia are safe."

Restrictions have since been imposed on three local government areas on the border in South Australia.

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