The Premier League pair are eyeing up a January deal for the Turkey international, according to talkSPORT.
The 23-year-old has started the new Super Lig campaign in excellent form, with the tricky wideman providing five assists for his side and scoring one goal so far in 2015-16.
This form has played a part in Besiktas’ rise to the top of the domestic table, with a win over Fenerbahce recently catching the eye.
Tore was in the Stamford Bridge youth ranks for two years but did not get a chance to prove his worth in the senior set-up for the current Premier League champions.
The Blues sold the German-born midfielder to Hamburg and he has also played for Russian side Rubin Kazan in his career to date.
talkSPORT claim that both Arsenal and Newcastle have been keeping a close eye on the Turk’s progress and want to tempt Besiktas with a January bid for Tore.
The Gunners did not sign any outfield players in the summer and as such it remains to be seen whether the usually reluctant Arsene Wenger is willing to strengthen mid-season.
Newcastle find themselves at the bottom of the Premier League table after eight games and as such are in desperate need of a morale boost if they are to avoid a relegation dogfight.