The 32-year-old's spell at the KC Stadium was cut short in 2011 when he was given a three-year ban after testing positive for the stimulant methylhexaneamine.
The suspension was halved after he assisted UK Anti-Doping with their investigation into the subsequent cover-up and he joined the Reds on a two-year deal when it expired at the start of this season.
Gleeson, who made his debut in Salford's win over Hull KR in February, was shortly afterwards handed the club captaincy by caretaker coach Alan Hunte but has missed the last three matches with a calf injury.
"I'm loving the captaincy, I'm just annoyed I'm not on the field at the moment," said the former St Helens and Warrington favourite. "It's just a few tweaks, stemming from one week playing on snow and the next on concrete, but hopefully I'll be back for Hull, that's the game I've targeted."
The Reds will give a debut to former St Helens prop Jake Emmitt, who joined them a week ago from Castleford in a straight exchange for Ryan Boyle.
"It was a pretty easy swap," Hunte said. "Ryan Boyle has been fantastic for this club but he's a Cas lad and Jake's a Warrington lad.
"We haven't lost anything player-wise and a fresh face might add something to the squad.
"Jake's a really good professional who has come through really good systems."
Hull are without Salford's former captain Daniel Holdsworth, who is on the sidelines for an indefinite spell after suffering the repeated effects of concussion, as well as their entire first-choice back row, with Danny Tickle joining Gareth Ellis and Ben Galea on the injury list.