According to journalist Fabrizio Romano, the decision on whether to accept now rests with Newcastle United, who have already lost Sandro Tonali this summer and could yet see Bruno Guimarães depart before the window shuts on 1 September, as Matthias Jaissle continues reshaping his squad.
Turkish outlets have reported conflicting figures on the size of Besiktas' proposal.
Spor Depor described an offer worth €7 million plus a further €5 million in add-ons, well short of the roughly €20 million valuation Newcastle have been holding out for, while Takvim reported that an "official offer" has since been submitted and the Turkish club is waiting on a response.
Willock, who started just ten of his 24 Premier League appearances last season, has also carried a groin issue since suffering it in pre-season against Strasbourg, though that has not altered Newcastle's willingness to sell.
Crystal Palace, West Ham and Leeds have all previously been credited with interest in the England-capped midfielder, though those clubs are understood to have cooled their pursuit, leaving Besiktas as the clear frontrunner.
The Turkish side have already spent heavily this summer, with their outlay understood to have reached around €67 million, including a reported €30 million fee for Benfica midfielder Orkun Kökçü. Willock has one year remaining on his Newcastle contract and has previously indicated he expects to stay on Tyneside this season, though his standing in Eddie Howe's midfield has slipped following the arrival of Anthony Elanga and the emergence of academy options.
Newcastle's central midfield picture remains in flux more broadly, with the Magpies having already sanctioned the sale of Tonali earlier in the window and Guimarães' long-term future still the subject of separate transfer speculation.