Does the time difference between Chicago and Moscow change during the year?

Yes. Because of daylight saving time the gap moves between −9h and −8h.

When is the best time to call Moscow from Chicago?

There is no overlap between standard working days, so one side has to call early or late.

Chicago is 8h behind Moscow. This gap is not fixed: daylight saving time moves it between −9h and −8h during the year.

Chicago

United States

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Moscow

Russia

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A 9-to-5 working day in Chicago does not overlap one in Moscow at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Chicago · United States	America/Chicago	UTC−5
Moscow · Russia	Europe/Moscow	UTC+3

What is the time difference between Chicago and Moscow?: Chicago is 8h behind Moscow. This gap is not fixed: daylight saving time moves it between −9h and −8h during the year.
Does the time difference between Chicago and Moscow change during the year?: Yes. Because of daylight saving time the gap moves between −9h and −8h.
When is the best time to call Moscow from Chicago?: There is no overlap between standard working days, so one side has to call early or late.