The Hoops were paired with the Northern Ireland champions in Monday's draw for the second qualifying round and were set to play the first leg at Celtic Park.
Both clubs are now awaiting a decision from UEFA after Cliftonville requested the opening fixture take place in Belfast.
A Celtic statement read: "The first leg of the tie was scheduled for Celtic Park on July 16/17, with the return match taking place a week later in Belfast on July 23/24.
"However, at the request of Cliftonville, there is a possibility of the matchdays for the tie being swapped.
"Both clubs are looking into this at the moment and are awaiting a decision from UEFA.
"Celtic would reiterate that supporters make no travel arrangements until these details have been finalised."
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