Cambodia has proposed the Hindu-style shrine, Preah Vihear, that straddles the common border, to be listed as a World Heritage site.
However, the bid reached a deadlock last year after Thailand viewed that the demarcation along the border should be settled first.
Thai Foreign Minister Noppadon Pattama said Virasakdi will hold talks with the Cambodian side to discuss the issue before a Unesco committee meets from July 4 to 12 in Canada.
If the problem cannot be settled, it could be politicised as the general election in Cambodia approaches.
"Thailand has lent its full support for Preah Vihear to be a World Heritage site," Virasakdi said, but added the demarcation in the area should be solved first.
Unesco wants both countries to discuss Cambodia's proposal to register Preah Vihear as a heritage site.
The ownership of Preah Vihear has been a source of contention between Thailand and Cambodia and the International Court of Justice ruled in 1962 that the temple belonged to Cambodia.