The 28-year-old only returned to the Premier League last summer after joining on loan from AC Milan on deadline day.
Forest have the option of turning his temporary stay into a permanent switch.
But the Reds are unlikely to activate their buy option, and could part ways with the Belgium international in January.
Origi has started only two Premier League games, and has failed to score in any of his nine top-flight appearances.
It is believed that Origi, who scored in Liverpool's 2019 Champions League final victory over Tottenham, is attracting interest from the MLS.
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The former Lille star arrived at the San Siro on a free transfer in July 2022 after his contract expired at Anfield.
Signed on a four-year deal, Origi flattered to deceive in his opening campaign in Italy having scored just twice in 27 Serie A matches.
Meanwhile, Forest are also open to offers for midfielder Orel Mangala after being charged with breaching Premier League Profit and Sustainability Rules earlier this week.
If found guilty, Forest could be hit with a fine or points deduction.
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PSR rules state that clubs can only lose a maximum of £105million over a three-year cycle, or £35million-a-season.
Since earning promotion to the Premier League in 2022, Forest have spent around £250m on 43 new signings since being promoted in 2022.
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In that period, they only made around £5m from sales.
Despite Forest selling forward Brennan Johnson to Tottenham last summer for £45m, that sale can't be included in these accounts and will be in the next three-year cycle.
In a statement released on the club's website, Forest confirmed they hope to find a resolution to this situation.
"Nottingham Forest acknowledges the statement from the Premier League confirming that the club has today been charged with a breach of the league's profitability and Sustainability Rules.
"The club intends to continue to cooperate fully with the Premier League on this matter and are confident of a speedy and fair resolution."
Forest will be sweating over a possible points deduction with the club sitting in 15th spot in the table - just four points above the bottom three.