The 29-year-old is among several players who could leave the Emirates Stadium before September's deadline, having fallen down Mikel Arteta's pecking order behind Kai Havertz and Viktor Gyokeres.
Jesus was left out of the squad entirely for Arsenal's Community Shield win over Manchester City, a decision Arteta described at the time as a "technical" one.
Napoli had been the most active suitors for Jesus in recent weeks, but according to Spanish outlet AS, negotiations between the forward and the Serie A club have stalled, with the Brazilian understood to be reluctant to move away from English football at this stage of his career.
That has opened the door for Premier League interest, with both Everton and Ipswich having made contact with Arsenal over a potential deal.
Everton manager David Moyes is working with Beto and Thierno Barry as his current centre-forward options, neither of whom has yet established himself as a prolific goalscorer at Premier League level, making further attacking reinforcement one of the club's priorities before the window closes.
Ipswich Town, newly promoted back to the top flight, have already added forward Emersonn to Gary O'Neil's squad this summer but could see Jesus as a further boost to their survival hopes.
Arsenal have set an asking price in the region of £18-20m for Jesus, who enters the final year of his contract next summer.
The forward has scored 79 Premier League goals in his career, three short of Roberto Firmino's record for a Brazilian player in the competition, and has been an important figure since joining from Manchester City in 2022. With his role increasingly reduced, though, both player and club appear open to a fresh chapter elsewhere in England before the deadline arrives.