The Black Eagles were reported to have entered talks with the former England international last week and head coach Şenol Güneş gave the green light for negotiations to continue today.
Güneş was appointed Beşiktaş manager on a two-year deal today and underlined that the team need strengthening in a number of positions.
The Istanbul giants struggled at right-back last season. Turkish international Serdar Kurtuluş did not have a replacement and Slaven Bilic was forced to play central midfielders Necip Uysal and Atiba Hutchinson as a fullback to fill the void when he was not available.
Johnson is expected to sign for Beşiktaş next week.
The 30-year-old right-back is currently available on a free transfer as Liverpool did not renew his contract.
Johnsonjoined Liverpool from Portsmouth for £10 million in 2009 and has made 200 appearances for the Reds.