Tsai Ing-wen, running for a second term, favours the status quo, and does not want closer ties with Beijing.
Meanwhile her main rival, Han Kuo-yu, promises to ease tensions with China.
Two years ago, Ms Tsai's party lost the local elections, but she is now ahead in the polls - which some observers attribute to the Hong Kong protests.
Ms Tsai has expressed support for the protesters, a popular stance with those who fear Taiwan being overtaken by mainland China.
Elections for Taiwan's legislature, where Ms Tsai's party hold a majority, are also taking place.
