In a speech to delegates at the opening of the UEFA Congress, Erdogan said: "I would like to repeat the request to host the European Championships in Istanbul in 2020."
The proposal comes as a blow to Istanbul's 2020 Olympic candidacy as bidding for two major events which are usually held only weeks apart is unlikely to be successful.
UEFA President Michel Platini questioned Turkey's ability to host both events in the same year.
"I heard what the prime minister said and we will see what countries present bids. Turkey had a strong bid in 2016 and I think they will be even stronger in 2020.
"Now the only small issue I have in respect to this is that in 2020 Istanbul is also a candidate for the Olympics so if they have the Olympics and the Euros it's maybe a bit of a handful," Platini told reporters after UEFA's congress in the Turkish city.
"If Istanbul does not get the Olympic Games the Turkish bid will be a stronger bid and one that I will support and one that I will vote for. Last time [for 2016] I did not vote, but this time I will vote for , because I believe Turkey should host a Euros."
The International Olympic Committee will elect the winning bid for the 2020 Games in September next year with Qatar's Doha, Madrid, Tokyo and Azerbaijan's Baku also in the running.
IMPOSSIBLE TASK
"Honestly, if Istanbul already has the Olympics before then I won't vote for Turkey because it is impossible to have the Olympic Games and the Euros in the same month in the same city. That's my position," Platini said.
Erdogan's announcement comes as Turkish footballing authorities grapple with a match-fixing investigation, under which 93 people including the chairman of champions Fenerbahce are being tried in an Istanbul court.
UEFA will formally start the bidding process next week when all 53 nations will be invited to apply for what is generally regarded as the world's third biggest sports event after the summer Olympic Games and the World Cup.
The tournament, first staged with four teams in 1960, will have 24 finalists from 2016 onwards which limits the number of countries who can stage it single-handedly.
Turkey lost out by a single vote to France in a bid to stage the 2016 finals.
Istanbul 2020 Games bid officials said the country's support for the Olympics remained guaranteed despite a likely Euro 2020 bid.
"Turkey's president, prime minister and sports minister have repeatedly reiterated that Istanbul 2020 is a strategic national priority, and a personal passion," bid officials told Reuters in an emailed statement.
"Every level of government and every party is aligned behind this bid: the Olympic and Paralympic Games are seen as a crucial stimulus for Turkey's national development strategy. Should Turkey bid for Euro 2020, that level of commitment will remain guaranteed," they said.
