The two councils are located just outside the Wanni war zone where security forces defeated Tamil Tiger rebels in mid-May.
Media rights groups have slammed the authorities for not allowing independent media access to the region.
"It is unacceptable that the government should impose such a ban (on independent journalists) on nothing more than the vaguest security grounds," Paris-based Reporters Without Borders said.
"As well as violating the population’s fundamental rights by preventing them from circulating freely, this measure dashes any hope of a transparent election."
Saturday's vote does not cover areas recently captured from the Tiger rebels but travelling to both areas