Sky Sport reports that the Turkish club's opening offer leaves the two sides some distance apart, with Napoli holding out for a fee closer to €10m-€12m given they must pass on 40% of any sale to Chelsea under the terms of Lukaku's original transfer, but there is a shared willingness to find an agreement in the coming days.
The Belgian striker did not travel with the squad to their training camp, a clear sign his time in Naples has run its course, with his agent, Federico Pastorello, spotted at the club's retreat as talks intensified.
Napoli have also opened concrete interest in Arsenal's Gabriel Jesus as a possible replacement, with Giovanni Branchini, Massimiliano Allegri's agent and the Brazilian's Italian representative, also present at Castel di Sangro, alongside André-Frank Zambo Anguissa's agent Maxime Nana, on a day of high-level meetings at the club's retreat.
Branchini spoke warmly of Allegri's start to life in Naples but declined to be drawn on the Gabriel Jesus speculation, saying he did not want to comment on simple market rumours.
Former Italy international Francesco Graziani, speaking to Radio Kiss Kiss Napoli, was blunt in his assessment of Lukaku's situation and cautioned that Napoli's negotiating position may be weaker than the club would like.
"On Lukaku there's no doubt, he'd already fallen out two months before the end of last season, there's no chance.
"The only thing is Napoli will try every way to get it done, since he still has a year left on his contract and they'd like to bring in something.
"But at this point Napoli don't set the price any more; if the player digs his heels in and tells you 'there or nowhere,' then you're in trouble. It weighs on a season's finances. Napoli have to hope to recover as much as possible, because by now Lukaku is gone, both in his head and in his feet," he said.
On the season ahead, Graziani struck a more optimistic note about the squad Allegri is assembling.
"If Napoli get the right pieces in place, I think they can fight for something more, why not the title... Inter, Juventus and Milan will fight for that goal too. It then depends on the kind of harmony this squad creates, with a new coach and new motivation. I don't put too much weight on summer friendlies, they only tell you so much. Although some indications do come through, for better or worse."