The Gunners snapped up centre back Gabriel Paulista from Villarreal in January, but will possibly look to add a younger defender to grow behind Laurent Koscielny and Per Mertesacker next season.
It would therefore make sense for them to land Gomez, who has made 23 appearances for Charlton in all competition this season despite being just 17.
The teenager has made 25 appearances for England across three age groups at youth level, and is regarded as one of the top young prospects in the Football League.
The 6ft 2in defender could now be on the move, with The Mirror suggesting that Arsenal lead Manchester City and Manchester United in the race for his signature - because of their relationship with Charlton.
Having signed Carl Jenkinson from the Addicks in 2011 and loaned Francis Coquelin to Charlton this season, there is certainly a history of dealings between the two clubs.
However, whilst Arsenal may be leading the race to sign Gomez, The Mirror also suggest that Charlton want £8m for the teenage defender - a huge amount of money for someone so young.
With stardom predicted, Gunners boss Arsene Wenger may be willing to take a risk and bank on him reaching his potential, as many young players have done previously under the Frenchman - but the asking price could complicate the race to sign Gomez this summer.