Johnson is currently a free agent after Liverpool decided not to renew his contract.
Beşiktaş could sign Johnson on a free transfer if they manage to come to an agreement.
Johnson was closely linked with West Ham but Slaven Bilic seems to be keen on extending Carl Jenkinson’s loan deal.
The Daily Star report also claims that Manchester City are interested in signing Johnson as he would count as a home grown player.
The City deal would however, come as a surprise considering Johnson wants regular first team football.
Beşiktaş would be able to provide Johnson with a lucrative contract and first team football but have so far been unable to reach a deal.
Despite the latest rumours Jonhon’s odds were slashed from 2.65 to 1.80 on Skybet as being favourites to sign the former England international over the summer transfer window.