According to Glenn Price of ESPN FC, the Toffees are interested in making the Belgium international the first player to transfer between the Merseyside rivals since Abel Xavier in 2002.
"A source said that Liverpool had become aware of potential interest from Everton in the 23-year-old but were yet to receive any contact from their local rivals," Price said. "Everton are keen to add more firepower to their ranks as recognised strikers Cenk Tosun, Dominic Calvert-Lewin and Oumar Niasse have just two Premier League goals between them so far this season."
James Pearce of the Liverpool Echo also reported that the Toffees have an interest in the player, although it's said he won't be allowed to move on the cheap.
According to Pearce, the Reds will ask for more than £20 million before they decide to sell the striker. It's added that Origi was poised to join Wolverhampton Wanderers in the summer, but after a £22 million fee was agreed between the two teams, the player turned down the move.
Neil Jones of Goal reported that the Reds forward was also wanted by Borussia Dortmund. However, Origi didn't want to go out on loan again following a spell at Wolfsburg:
Pearce added that Liverpool are expecting Wolves to make another move for the forward in January.
This season, Origi has found it impossible to get game time for the Reds. Not only does he have a front three of Mohamed Salah, Roberto Firmino and Sadio Mane to dislodge, but manager Jurgen Klopp can also call upon a rejuvenated Daniel Sturridge and new signing Xherdan Shaqiri.
Per journalist Jack Lusby, Klopp has commented on the Belgium international's condition, although he's yet to play a competitive game this season:
However, during his stint at Wolfsburg in 2017-18, he looked well below the standards required and spurned a number of big chances:
Origi had a reputation as one of the game's brightest prospects when he joined Liverpool from Lille following the 2014 FIFA World Cup, and while there have been some high points for him at Anfield—he enjoyed a particularly prosperous 2016—overall, he's underwhelmed and been inconsistent.
Everton may lack natural options up top, but there have been signs that Richarlison can lead the line—he has four goals this season—while Gylfi Sigurdsson has also chipped in with five goals in all competitions.
Even so, it would be a big surprise if the Reds did opt to sell a talented player, albeit one on the fringes of the first team, to their neighbours.