Jennings will link-up with his new team-mates on Friday ahead of Saturday's League One clash with Coventry at stadium:mk.
The highly-rated 20-year-old, who spent two seasons with German giants Bayern Munich before signing for Barnsley, will be available for the Dons' league games against the Sky Blues, Wolves, Port Vale, Crawley, Brentford, Colchester and Crewe.
Jennings began his career as a schoolboy with Liverpool but after failing to break through into the Reds' first team he joined Tranmere, making his debut for the Wirral outfit aged 18 in September 2010.
He finished his debut season with six goals and 32 appearances - form which earned him a £600,000 move to Bayern Munich.
In June this year, Jennings signed a three-year deal with Championship club Barnsley but the tricky winger has been limited to just five appearances for the Tykes.