The 28-year old midfielder is a free agent after departing Middlesbrough at the end of January and has been training with Hibs who he started his career with back in 2003.
Reports have suggested that Fenlon was set to sign Thomson for the rest of the season but he has played that talk down by making it clear the ex-Rangers man is out of the Edinburgh clubs price range.
"Kevin is just training with us. We don't have any money to do anything, so it's not as if I'm pushing for anything," Fenlon said.
"I spoke to Kevin, he was looking for training facilities and that is all there is at the moment - although in football you never know.
"Even reaching the Cup semi-final wouldn't free up any more cash.
"In fairness to the club, when I took the job I knew where we were. We lost money last year and it's difficult to get the balance right."