Erdogan, himself a former footballer, met Platini at the ruling AK Party's headquarters in Ankara on Monday ahead of the UEFA Executive Committee meeting in Istanbul on Tuesday and Wednesday.
The UEFA Congress of Europe's 53 national associations will be held in Istanbul on Thursday. Host venues, including stadiums for the 2014 UEFA Champions League and Europa League finals, will be appointed at the Executive Committee meeting.
The allegations emerged last July when police detained those accused of involvement in rigging 13 matches, including Fenerbahce's 4-3 victory over Sivasspor which clinched the league championship on the final day of last season.
Radikal newspaper said Platini had told Erdogan in their talks that it would be a good idea for Fenerbahce to withdraw a 45-million-euro case which the club has opened against UEFA at the Court of Arbitration for Sport (CAS) over its exclusion from this season's Champions League.
Monday's talks were also attended by Turkey's sports minister, UEFA's Turkish vice president Senes Erzik and the new chairman of the Turkish Football Federation (TFF), Yildirim Demiroren.
