Akinbiyi has become a fringe player at Turf Moor this season and approached the Gills earlier this season with a view to a return to a club where he is still held in high regard by the club's fans.
And Stimson said: "I would welcome someone like him with open arms because he's a fantastic footballer. You don't know. When the window opens for a loan deal for four weeks we would love to take him.
"Financially it would be way, way out of our bracket but if the chairman could pull something like that off for a month or two, then it would definitely be worth it.
"We haven't used a lot of the money from the Aston Villa game so we are just waiting around for a crazy deal like that to happen."