Jeffers, 24, had featured in the Clarets' last two pre-season matches and impressed the Clarets boss Rod Stringer, especially in City's 2-1 win against a Southend United XI.
The forward began his career at Coventry City and made six first team appearances for the Sky Blues, before having spells at Cheltenham Town, Cambridge United, Peterborough United and Newport County.
Last season Jeffers made over 30 appearances in League Two for Yeovil Town before his release in the summer, with the striker also having two England u19 appearances under his belt.
Jeffers will now go straight into contention for selection against Truro City on Saturday as the Clarets open up their league campaign at Treyew Road.
Updated 15:29 - 3 Aug 2016 by Scott Hatley