Allen, who has played with New England since his rookie year of 2013, is coming off a 2018 campaign in which he averaged 45.1 yards per 64 punts, which was right along with his terrific career average of 45.3 yards a punt.
In 409 career punts, Allen only had two punts blocked.
The 29-year-old Allen was in the last season of his current contract and set to make $900,000.
Rookie Jake Bailey, who the Patriots drafted in the fifth round, is now likely to take on the team's punting duties. In the Patriots' two preseason games thus far, Allen had five punts with a 43.8 average, a long of 57 and two inside the 20-yard-line. Bailey has posted an impressive 49.5 average with a long of 54, but it's come on just two punts.