Rogic will link up with the green and gold on December 27 meaning he'll miss the Hoops derby match against Rangers on the 29th.
With Australia not kicking off their Asian Cup campaign in the UAE until January 6 against Jordan, Rodgers believes the Socceroos should be allowing Rogic to spend more time with his club before the month-long tournament.
"At this moment in time he's due to play his last game for us away at Aberdeen and then he's due to fly out to meet up with the Australian squad on the 27th so that would mean him missing the last game," Rodgers told BBC Scotland.
"That's not something we're overly happy about because we want to work in a relationship with the federations.
"I spoke with Graham Arnold, the manager, here and I had two really good days of trying to ensure relations are good between international teams and club teams.
"But you do that through working together and what is the best for each player. And we feel Tom can play and be available for the 29th and fly out after the game and meet up in Dubai on the 30th and still be there virtually a week before he's due to play his first game.
"Tom wants to play in the game but he also wants to play for Australia. It's a position the player doesn't need to be in.
"It's disappointing if we can't find a solution but I'm hopeful that they can see sense with it and see that they could still get their players in good preparation time and be ready for the tournament's start."
If Rodgers thinks he has it tough, Hibernian boss Neil Lennon is set to lose three Australians with Jamie Maclaren, Mark Milligan and Martin Boyle all called up for Socceroos duty.
While yet to be officially confirmed, the green and gold are expected to play a friendly against Oman on December 30.