Houston currently has Trevor Daniel in place as their punter for the 2019 season.
Allen, 29, wound up signing on with the Patriots as an undrafted free agent out of Louisiana Tech back in 2013. Allen had one year remaining on his rookie contract when he and the Patriots agreed to a new three-year extension back in 2015.
Allen re-signed with the Patriots back in March only to cut him loose a few weeks ago.
In 2018, Allen appeared in all 16 games for the Patriots and totaled 2,885 yards on 64 attempts (45.1 YPA) to go along with 21 punts down inside the 20-yard-line and a long of 66 yards.
Wile, 27, originally signed on with the Panthers as an undrafted free agent out of Michigan back in 2015. He bounced on and off their roster until signing with the Cowboys during the 2016 offseason.
Wile lasted six months in Dallas before having brief stints with the Falcon, Cardinals and Steelers. The Vikings claimed him off of waivers at the start of last year before cutting him loose last week.
In 2018, Wile totaled 3,808 punt yards on 84 attempts (45.3 percent) over the course of 16 games.