He has played in 96 NFL games and has a career average of 40.2 yards per punt.
Allen will have to beat out punter Sam Irwin-Hill, who signed Feb. 7.
Bosher, a sixth-round pick in 2011 out of Miami, appears headed to free agency after punting for the Falcons for the past nine seasons.
The Falcons also had problems at the kicker position and signed Koo on Oct. 29 after releasing long-time kicker Matt Bryant.
Koo, 25, a former Georgia Southern kicker, made 23 of 26 field-goal attempts and 15 of 16 extra-point tries.