Chen Shui-bian said on Tuesday that he would nominate a new prime minister before the Lunar New Year, which starts on 29 January.
"I have accepted his resignation. He has stepped down after accomplishing his job," Chen told reporters in Taiwan's island of Quemoy, where he was visiting troops.
Hsieh, 59, told a news conference in Taipei that he would formally resign with the entire cabinet next Monday.
Under Taiwan's political system, the democratically elected president appoints the prime minister who forms the cabinet and runs day-to-day government.
Taiwanese media have speculated that Chen will appoint Su Tseng-chang, his popular former chief of staff, in an attempt to shore up his own poor approval rating.
Chen is struggling against criticism that he has become a lame duck after his pro-independence Democratic Progressive Party (DPP) was defeated by the opposition Nationalist Party (Kuomintang or KMT) in polls for local government chiefs on 3 December.
Hsieh is viewed as a moderate in the DPP and another presidential contender, against Su and Ma Ying-jeou, the KMT leader.
But his image has been tarnished by a corruption inquiry into a subway project in Kaohsiung, where Hsieh was formerly the mayor.
The KMT and its allies oppose Taiwanese independence and favour closer ties with China.