Gleeson, 32, has been out of the game since failing a drugs test while playing for Hull in 2011, but his subsequent 18 month ban is up next month, enabling him to return to action.
He is being lined up to replace Australian Moon, who this week left the City Reds to pen a three-year deal with Super League champions Leeds.
According to local press, the move will be announced when Salford’s protracted takeover by Peel Holdings is completed.
Gleeson, who has earned 26 international caps during his career, the last coming at the 2008 World Cup, has made more than 300 first grade appearances since making his Super League debut for Huddersfield in 1999.
He moved on to St Helens in 2001, spending three seasons at Knowsley Road, during which time he was banned for four months for his involvement in a betting scandal.
The centre signed for Warrington on completing his ban, and spent four and a half years at the Halliwell Jones Stadium before signing for hometown club Wigan midway through 2009, but moved on to Hull in 2011 following disciplinary problems and a “serious distraction in his personal life”.
In June 2011, after making just seven appearances for the Airlie Birds, he was granted leave for a “stress related illness” but was later found to have failed a drugs test after a game against Salford in May.
He was handed a three-year ban, with half being suspended, dating from June 12th 2011, which was the date of his last rugby league appearance, against Harlequins.