Metro reports that Rodgers has turned his attention to the 21-year-old after losing out on Henrikh Mkhitaryan, who join Borussia Dortmund last week.
Shaqiri has failed to become a regular at Bayern Munich since joining the club from FC Basel last summer.
The Switzerland international is expected to hold talks over his future at Bayern with new boss Pep Guardiola next week.
Reports suggest Shaqiri could ask for a move away from Bayern Munich in order to get regular match time.
Brendan Rodgers is reportedly willing to pay up to £20million for the highly-rated Swiss youngster.
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