"I don't think Fleetwood have got anything to be afraid of," he said.
"The likes of Crawley and Stevenage have gone up and then up again. I don't see why we can't because the club are making some good signings."
The striker continued: "The lads who are already here have done magnificently well to win the Conference like they did. There are some good players here."
Gillespie, who scored 12 goals in League One for Colchester last season, admitted his transfer came as a surprise to him, but once he learned of Fleetwood's interest, he was excited at the prospect of a move to Highbury.
"I was fully planning on playing for Colchester United next season," added the former Bristol City and Cheltenham forward. "I wasn't even thinking of a move. It came a little out of the blue.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.