Tshibola, 20, came through the Royals' academy system and signed his first professional contract last summer.
He made his professional debut against Nottingham Forest in August.
Bingham, 21, joined Mansfield from Wigan Athletic in the summer and has scored four goals in 20 games for the Stags in all competitions.
Pools are awaiting clearance from the Football Association and the Football League for the duo to feature against Wycombe Wanderers on Saturday.
Hartlepool United have signed Reading midfielder Aaron Tshibola on a one-month loan deal and Mansfield striker Rakish Bingham on a season-long loan.