The 22-year-old will look to earn a longer deal after training with the Cobblers as part of his recovery from a serious knee injury.
Former Swindon Town and Torquay United man Bodin knows Northampton assistant manager Alan Knill from his time with the Devon club.
His signing completes another busy week of deals at Sixfields as Chris Wilder completes his January reshuffle.
Ricky Holmes and Tom Newey both signed permanent contracts with the club on Wednesday after impressing on loan.