The 28-year-old is a free agent after leaving Middlesbrough earlier this year and he is now hoping to rejoin the club he captained after coming through the youth ranks at Easter Road.
Thomson could even be signed in time to feature in the Edinburgh derby against Hearts on Sunday.
"Kevin has been training with us and has indicated it is a place he would like to come and play and we are working on that," said Fenlon.
"Whether that is done for the weekend I am not sure. He has been training for two weeks, and looks fit but being outside the window it takes a lot longer."