21-year-old Fleetwood has been prolific this season netting 23 times in 23 games for the Green since he joined from Hereford United during the summer.
The Robins hope that they accept a loan request before the Thursday loan dealine and they hope to complete the signing the player in January with a fee muted between £100-200k.
Downing told www.thisisgloucestershire.co.uk "We have enquired about Fleetwood. He is prolific, he has done really well this season and you have to stand up and look at his figures.
"I am sure other clubs are interested in him too, but I am hoping to get someone in to make it more competitive here and freshen it up.