Both the Liverpool Echo and the Daily Telegraph are reporting the Reds will make a bid for the promising youngster of around £3.5 million but the Mail suggests the Addicks will be after a deal that could rise to £9 million.
Liverpool have reportedly been watching the defender - who can play at both centre-back and right-back - for the past 12 months.
He made a total of 24 appearances for Charlton's first team last season, and signing Gomez would fit in with a transfer policy that has been undertaken by the club's owners Fenway Sports Group since their arrival at the club, where they try and buy younger player and develop them.
It is said in the same reports that Sebastian Coates is set to leave Liverpool for around £4 million, and join Sunderland on a permanent deal after he spent the season on loan at the Stadium of Light - impressing towards the back end of the campaign as they avoided relegation from the Premier League.
One does wonder whether Gomez would get sufficient game time to develop at the club given the experienced Kolo Toure and youngster Andre Wisdom have both signed new deals at the club - and there is still the case of Tiago Ilori who was signed in 2013 and has yet to make a first team appearance for Liverpool - instead spending time on loan at Granada and Bordeaux.
It makes you think his best chance of getting any real first team game time will come in the Europa League when the Reds begin their campaign in Europe, and he could hope to make an impression that way. However, given what Liverpool have done with young defenders in recent years, it wouldn't be surprising to see Gomez given a loan spell either at Charlton or elsewhere if the Reds complete the deal.