The midfielder is set for a medical in London on Monday after the clubs agreed a deal worth in the region of £10.5m.
Palace were initially keen on bringing Gallagher back to Selhurst Park but the England international is eager to stay at Chelsea.
Ahamada has started 17 of Stuttgart’s 18 Bundesliga games this season and leaves the club during a relegation battle.
The 20-year-old began his career in the French lower leagues before joining Juventus and spent three years with the Turin giants.
Speaking of Ahamada’s move, Labbadia told Sky Germany: “We have to accept things that are not ideal for the sporting situation.”
Palace are plotting a move for a new right-back before Tuesday’s transfer deadline, with Galatasaray’s Sacha Boey having emerged as an alternative target to Aaron Wan-Bissaka, who now seems destined to stay at Manchester United.