Awoniyi has been out on loan since joining Liverpool in 2015 due to work-permit issues.
The 23-year-old is Nigerian and would command a fee in the range of £9 million.
And the Daily Star claims Awoniyi seen as a possible replacement for Odsonne Edouard.
Edouard's departure has become more likely after Celtic were dumped out of the Champions League.
The Bhoys lost 2-1 to Ferencváros on Tuesday.