The great rivals will meet for the first time in European competition when United make the short journey to Anfield next week, with the return leg the following Thursday.
And United stated in an e-mail that fans making the trip will receive a complimentary jersey, should they wish to claim one, in a bid to "generate the best possible atmosphere at Anfield".
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United will start delivering the shirts from the beginning of next week.
Sunderland were involved in a similar promotion last week when they offered free away shirts for the club's Premier League trip to West Ham.