The Argentine has played for a host of Football League teams, but he has been without an employer since he left Millwall earlier this year.
However, the former Swansea City left-back has now been given the chance to earn himself a permanent contract at Gresty Road.
"We're having a look for a few days. He's a free agent, although he's been back training with Swansea," Davis told BBC Radio Stoke.
Bessone, 30, started his youth career with Barcelona, before having spells with Swansea, Leeds United, Charlton Athletic, Swindon Town and Oldham Athletic.