Graham Rae claims that a decision was made on the match’s start time ahead of a meeting in Glasgow to discuss the fixture.
The Caley Jags chief concedes that the time is almost certainly down to TV companies, but added: “I don’t know how much say we will have, particularly as the minnows among the four clubs in the semi-finals.
“We want the best set of circumstances for our supporters and our club.”
Describing the decision to announce details before the meeting as “odd”, Rae told The Sun: “It just seems odd to me that we are being invited down for a meeting where everybody is represented — the fire [service], police, the clubs, the media — then they are announcing things ahead of time.
“I guess they want to get things sewn up before we meet, but it just seems odd.”
Celtic play Aberdeen in the second semi-final on Sunday April 14, with a 2pm kick-off. Both matches will be televised by Premier Sports, with BBC Scotland also screening the Hearts-Inverness meeting.