The move came after the Turkish Football Federation (TFF) rejected Fenerbahce's appeal on Thursday against their exclusion from this season's Champions League, in which it has been replaced by Trabzonspor.
More than 30 players and officials have been jailed pending trial over the match-fixing claims, including Fenerbahçe chairman Aziz Yıldırım and the coach and deputy chairman of Beşiktaş, in connection with alleged manipulation in 19 matches.
"Our deputy chairman Ali Koç and management board member Ali Yıldırım went to the Federation and conveyed our request to play in a lower league," Fenerbahçe said in a statement on its website.
Ali Koç said he expected the federation to announce their decision on the subject at the start of next week.
Fenerbahçe has said its loss of revenue due to the Champions League exclusion would be some 25 million Euros ($36 million), sending its shares tumbling 18 percent on Thursday. The shares rose 3 percent on Friday.

The club has said UEFA's stance on the issue is unlawful and that it could take legal action against both it and the Turkish federation, saying it was "innocent until proven guilty".

The federation said before the Champions League ruling that it would not take decisions regarding the clubs linked to the trial until the court had accepted the indictment and lifted a secrecy rule
That could take several months. However, any decision to relegate a club once the season ha d begun would mean it would be in a lower division in the following season, lengthening the punishment period.

UEFA had written to the federation on Tuesday saying Fenerbahçe must withdraw from the Champions League or the TFF should take the decision to bar them or face a disciplinary investigation, the federation said.Player Emre Belezoglu accused UEFA of unequal treatment. "There is no way UEFA could take such a decision against clubs such as Milan, Chelsea or Real Madrid. UEFA took this decision because we are a Turkish team, the decision is scandalous," he told a news conference on Friday.
TFF Chairman Mehmet Ali Aydınlar said UEFA warned that Turkey generally faced a punishment of up to eight years if the federation did not act.
The TFF, which took the decision to bar Fenerbahçe at an extraordinary board meeting on Wednesday before the Champions League draw, has postponed the start of the season by a month until Sept. 9 so its ethics committee can investigate the match-fixing claims.
Among games reportedly under investigation are Fenerbahçe's 4-3 victory over Sivasspor which clinched the title on the last day of the season and the cup final between Beşiktaş and IBB. Beşiktaş have returned the trophy pending the probe's outcome.