Hibernian and Hearts say the Scottish Professional Football League has compromised "sporting integrity" after promotion rivals Rangers' final game was moved for television coverage.
The Glasgow side will play Championship winners Hearts at Tynecastle on 3 May.
Hibs and promotion rivals Queen of the South and Falkirk will play on the Saturday, a day earlier.
However, the SPFL has hit back saying other clubs have the option of moving their own fixtures to Sunday too.
With Hearts having already won the title, the four clubs below are chasing three places in the promotion play-offs.
Hibernian chief executive Leeann Dempster said in a club website statement: "In previous seasons, the league has ensured that all of the final-round fixtures are played at the same time.
"I have been in contact with the SPFL following its announcement that one final-round fixture would kick off after all the other results are known.
"I have reminded them in very clear terms that - as the organiser of the competition - the responsibility for protecting its unimpeachable sporting integrity lies with them."
Her counterpart at Tynecastle, Ann Budge, backed up their city rivals' concerns.
"When it also became apparent to us that our fixture was the only game to be moved, we were equally alarmed as, rightly or wrongly, this casts a shadow over the sporting integrity of the Championship," she said on Hearts' website.
"While it may not be enshrined in the newly merged rule book of the SPFL, it is protocol the football world over for fixtures such as the last day of the championship to be played simultaneously.
"To move only one (or even two or four) out of five fixtures would seem to be leaving our game wide open to unsavoury accusations.
"I will be sending in a formal complaint to the SPFL in relation to this appallingly late decision."
While regretting any inconvenience to fans, an SPFL spokesman suggested that the circumstances "are in to way unusual".
"On this weekend, the Hearts v Rangers match is scheduled for Sunday 3 May to be broadcast on television and it is not clear to us why this has caused so much surprise," he said.
"This is a key game in the Championship season and will inevitably attract huge interest from fans around the world.
"It is unreasonable to expect all Championship fixtures that weekend to move to the Sunday when a number of them will be unaffected by the Hearts v Rangers tie.
"However, as a result of this fixture, all Championship clubs have been invited to submit a request should they wish to move games to the Sunday.
"To date, the only feedback we have received is from one Championship club, which has asked for its match to remain scheduled for the Saturday.
"For the avoidance of doubt, we have had no request from Hibernian FC to move its fixture to the Sunday."