Both had to be substituted with injuries in the first half of Sunday's 2-1 home defeat by Swansea City.
Bosnian Besic strained a hamstring and Mirallas hurt a knee, Martinez told reporters.
"I expect a bit of a period where they will be unavailable," he said.
"It is a real shame because both players gave us a really refreshing look and reached good levels at this stage of the season."
England defender Phil Jagielka could play after being rested against Swansea.
"We have five games in 13 days," Martinez added. "Phil Jagielka has been out for a long, long time and then he came back last week and played three games in seven days.
"The squad is ready to cope with players missing and, clearly, Phil Jagielka will be refreshed for what is ahead."
Everton lead 2-1 going into the second leg in a competition they have never won.
(Reporting by Steve Tongue, editing by Pritha Sarkar)
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