Jennings is a free agent after leaving Barnsley earlier this summer, cutting short his stay at Oakwell a year early.
The 22-year-old joined Barnsley in 2013 from Bayern Munich for a fee of £250,000, but saw his time in South Yorkshire hampered by injury.
Jennings made the surprise move to Bayern Munich in 2011 after catching the eye of the German giants at Tranmere Rovers.
The skilful wide-man never made a first-team appearance for Bayern, but he was a regular in the club's second side in the German fourth tier before returning to the UK in 2013 for personal reasons.
Jennings enjoyed a successful spell on loan at MK Dons in November 2013, making six appearances before being recalled by Barnsley.
MK Dons boss Karl Robinson is a big fan of the former Liverpool youngster and he feels he could prove a valuable addition to his squad as they prepare for their first ever campaign in England's second tier.