Ali Ihsan Yavuz, a ruling party deputy chairman, announced the decision hours after Turkey's top electoral authority rejected a request for a full recount of the votes cast in Istanbul.
Yavuz told reporters that the vote in Turkey's financial and commercial hub was marred by "organized irregularities."
"We will say that there were events and situations that clearly affected the result of the election and that we are using our right for an extraordinary appeal (to) demand a renewal of the election in Istanbul," Yavuz said.
Erdogan's Justice and Development Party, or AKP, suffered a major setback in the country's March 31 local elections. Opposition candidates won in Turkey's capital, Ankara, and squeezed out the ruling party in Istanbul, where Erdogan rose to power as mayor and which has been governed by his party and its Islamist predecessor for the last 25 years.
The opposition's mayoral hopeful, Ekrem Imamoglu, criticized the ruling party's plans and again called on the party to accept the results, saying Istanbul had no time to lose.
"The winner of the election is clear and the people on the street have accepted it," he said.
Earlier, the Supreme Electoral Board rejected the AKP's demand for a full recount in Istanbul but allowed a recount of 51 ballot boxes in the city's 21 districts, Recep Ozel, a member of the board from the ruling party, told reporters early on Tuesday.
Separately, the party is seeking a repeat of the vote in the district of Buyukcekmece, as it says thousands of people were wrongly registered.
On Tuesday, police were carrying out inspections at addresses around the Buyukcekmece district, the state-run Anadolu Agency reported.
The opposition has accused the party of being a "bad loser" and has repeatedly called on Erdogan to concede that his party lost.