The News understands former AFC Bournemouth and Burnley midfielder Stock will become Bradbury's assistant manager at the National League Spitfires.
Most recently boss of Weymouth, three-cap Wales international Stock, 40, was relieved of his duties by the Terras in January having spent two years in charge – after a run of seven successive league defeats.
Former Pompey and Manchester City striker Bradbury – appointed Eastleigh boss on Monday – knows Stock well having signed him in 2014 while in charge of National League South Hawks.
Stock stayed at Westleigh Park for five years, before retiring in 2019 - with Bradbury his boss for the whole of that period.
The pair enjoyed some great times at the Hawks as both were part of the club reaching the National League for the first time following promotion from the sixth-tier in the 2017-18 season.
Bradbury, who left his role as Crawley assistant manager to become Eastleigh boss, will keep Jason Bristow on his first-team coaching staff, following his spell in caretaker charge prior to the 46-year-old's arrival at the Silverlake Stadium.
And Stock is expected to join him as assistant manager as Bradbury looks to assemble his coaching staff. Bradbury's Eastleigh host Wealdstone in the National League tomorrow for what will be his first fixture in charge.