Three Italian clubs have contacted Origi's representatives about a pre-contract agreement.
Origi would prefer to stay in England, meaning either a club would have to pay him this month or take him as a free agent in the summer.
The Belgium international may not make a decision until the summer.
Origi's contract could be extended by a year if he makes at least one start in the premier league this season.
Origi is described as one of the best finishers I have ever seen by Jrgen.
It is thought that he would cost less than $10m if he were to be sold.
Origi has been close to leaving the club before, and even spent a season on loan in Germany at Wolfsburg, but has yet to find a suitable club.
After Origi came off the bench to score a last-minute winner for the Reds at Wolves last month, he was praised by his boss, who said he would be happy for the forward to find a new club.
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There are highlights from the win against Wolves.
"Origi is an incredible forward and for different reasons he did not play that often, but I hope one day he finds a manager that plays him more than I do," he said after Origi made his 100th appearance.
He's one of the best runners I've ever seen. He doesn't play all the time but he is a positive boy, loves the club, and he did an amazing job for us.
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