The Welsh defender scored his first Everton goal on Tuesday, to help Ronald Koeman's side to a much-needed win at home to Arsenal.
The 32-year-old has often been linked with a move to the Emirates over the years, before sealing his summer switch from Swansea for around £12m.
It was his 17th appearance for the Merseysiders and Williams' first club goal since March - which was a winner against Arsenal at the Emirates.
Before signing Shkodran Mustafi from Valencia in the last transfer window, Arsene Wenger's side have been in need of a central defender, with many suggesting Williams was the answer.