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Fenerbahce's appeal on Thursday against their exclusion from this season's Champions League, in which they have been replaced by Trabzonspor.
More than 30 players and officials have been jailed pending trial over the match-fixing claims, including Fenerbahce chairman Aziz Yildirim and the coach and deputy chairman of Besiktas,
"Our deputy chairman Ali Koc and management board member Ali Yildirim went to the Federation and conveyed our request to play in a lower league," Fenerbahce said in a statement on their website.
Koc said he expected the federation to announce their decision at the start of next week.
The federation said before the Champions League ruling that it would not take decisions regarding th e 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.
REVENUE LOSS
Fenerbahce have said their loss of revenue due to the Champions League exclusion would be some 25 million euros, sending their shares tumbling 18 percent on Thursday. The shares rose 0.5 percent on Friday.
The club have said UEFA's stance on the issue is "unlawful" and that they could take legal action against both UEFA and the Turkish federation, saying they were "innocent until proven guilty."
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.
UEFA had written to the federation on Tuesday saying Fenerbahce had to withdraw from the Champions League or the TFF should take the decision to bar them or face a disciplinary investigation, the federation said.
TFF Chairman Mehmet Ali Aydinlar said UEFA warned that Turkey generally faced a punishment of up to eight years if the federation did not act, Turkish media reported.
The TFF, which took the decision to bar Fenerbahce at an extraordinary board meeting on Wednesday before the Champions League draw, has postponed the start of the season by a month until September 9 so its ethics committee can investigate the match-fixing claims which allegedly involve 19 matches.
Among games reportedly under investigation are Fenerbahce's 4-3 victory over Sivasspor which clinched the title on the last day of the season and the cup final between Besiktas and IBB.
Besiktas have returned the trophy pending the probe's outcome.
