The 34-year-old winger ended last season at Bradford after being released by Sheffield United.
The Belfast club were contacted by Gillespie's agent Phil Munnelly and the matter has now be referred to chairman Aubry Ralph.
Manager Alan McDonald told the Belfast Telegraph: "Of course it is interesting but I have passed it above my head.
"It is well documented the position the club is in, in terms of finances and the budget being cut. There are quite a few things that need to be weighed up."