The former UEFA Cup winners are ninth in the Turkish championship after eight games, eight points adrift of leaders Bursaspor. Galatasaray also failed to reach the group stage of this season's Champions League or Europa League.
It was not immediately clear who would take over from Dutchman Rijkaard but local media said the 17-times Turkish champions had made an official proposal to former Galatasaray and Turkey coach Fatih Terim.
Galatasaray held a management board meeting this week after Saturday's 4-2 home defeat by Ankaragucu to discuss the situation and said at the time it would announce decisions during the week.
"As a result of mutual talks with technical director Frank Rijkaard... we have decided to part ways as of October 20," the club said in a statement.
Trainer Johan Neeskens and their assistants had also left the club, the statement said.
"We have a debt of gratitude to the esteemed Frank Rijkaard for his work and efforts for Galatasaray," it added, without disclosing a reason for the decision.
The Dutchman took the helm at the Istanbul club in 2009 after five seasons in charge of Barcelona.
Galatasaray took the decision ahead of a key derby match against Istanbul rivals Fenerbahce on Sunday.
The move also comes at a time of transition for the club, which is due to start playing next year at the new 53,000-capacity Turk Telekom Arena stadium. That will mark the end of a long era at the decrepit Ali Sami Yen stadium.
Turkish media said the club had made an official offer to Fatih Terim, who had two previous spells with Galatasaray in the late 1990s and between 2002-2004.
He subsequently coached the national side, taking them to the semi-finals of Euro 2008, but resigned after failing to qualify for the 2010 World Cup.
Sabah newspaper said Terim was expected to make his decision on Wednesday and said the club had made an offer to its former striker Hakan Sukur to take the post of sporting director.
Local media also listed Gheorghe Hagi, Tugay Kerimoglu, Graeme Souness and Turkish coach Hikmet Karaman as possible candidates to take over from Rijkaard.