Sky Sport Italia, La Gazzetta dello Sport and Calciomercato.com, the bid is €15m plus another €7m in various bonuses, taking the total to €22m.
Sportitalia have a slightly different version, €16m plus €4-5m in add-ons.
Fenerbahce had seemingly improved upon that last night, but it’s reported Zaniolo had already made up his mind to join Galatasaray.
That is probably because the team already includes several former Serie A players, including Dries Mertens, Mauro Icardi and Lucas Torreira.
The contract will be worth €3.6m per season including performance-related bonuses.
Roma were hoping to spark a bidding war between the Turkish teams ahead of their transfer deadline on Wednesday.
As part of their original deal to bring Zaniolo in from Inter, the Nerazzurri are owed 15 per cent of the sell-on fee.
The other important element that Zaniolo insisted on was a release clause allowing him to return to Serie A next season.
He was pushing for €25-30m, but it seems Galatasaray have managed to hold firm at €35m.