The two Premier League giants have just enjoyed productive periods during the summer transfer window, Arsenal's business seemingly being more specific to their London rivals who acted far later in August and September.
As it stands, the Gunners are benefitting from that activity, currently holding a position at the top of the Premier League standings, while the Blues are playing catch-up under newly-appointed head coach Graham Potter.
Nevertheless, according to Goal, both teams are still stringently assessing their options when it comes to players for the future.
The report claims that the two English sides have taken an interest in signing 19-year-old defender Renan should an opportunity arise.
At a time when Corinthians currently sit in third position in the Brasileiro Serie A standings, Renan has been provided with four starts in the league.
In his first three outings between June and August, Renan failed to help Corinthians to maximum points, with just one point being collected just games against Santos, Fluminense and Fortaleza.
However, Renan played the full 90 minutes on October 9 as Vitor Pereira's side recorded a 2-1 victory over Athletico Paranaense, helping establish a six-point advantage over the sixth-placed side.
With Renan possessing a contract until the end of 2024, Corinthians may not be in any rush to part ways with one of their up-and-coming players whose value could increase over the next 12 months.
Nevertheless, the report claims that Arsenal, Chelsea and Russian club Zenit St Petersburg have already spoken to Renan's representatives.
Those clubs are said to have learned that Renan has a release clause in his contract worth €50m (£43.22m), a value which is unlikely to be met at this current stage.
There is also an option, according to Transfermrket, where his deal can be extended by a further three years.