With nine days left before the transfer window closes, the Brazilian remains the club's preferred option to add goals and quality in the final third, an area that has caused problems over the past two seasons.
Jesus finds himself increasingly on the fringes at Arsenal, having been left out of the squad entirely for both the Community Shield defeat to Manchester City and the Premier League opener against Coventry.
Despite that, Napoli's hierarchy cannot push through a deal until at least one permanent sale is completed first.
Sporting director Giovanni Manna confirmed the need to sell before Napoli's match against Genoa, with Noa Lang among the players attracting the most interest. Atalanta submitted an offer worth €2 million plus €1 million in bonuses for a loan with a €20 million option to buy, but Napoli rejected the proposal, insisting they will only consider permanent departures.
The Dutch winger nonetheless appears to remain part of Allegri's plans regardless of the interest, much like Lorenzo Lucca, whose place in the squad looks increasingly secure following a strong display in Genoa, where his flicked assist set up Antonio Vergara's goal.
Napoli are also exploring alternatives to Jesus should that move not materialise, with Marc Guiu of Chelsea and Manchester United's Joshua Zirkzee both under consideration.
Jesus remains the priority, however, given his experience at the highest level and versatility across the front line, qualities that have made him particularly appealing to Allegri as Napoli look to add firepower before the market closes.