Brian Costello of the New York Post added that the deal could be worth up to $3 million.
Siemian is expected to serve as the backup to 2018 third overall pick Sam Darnold.
Siemian is a former seventh-round pick out of Northwestern who spent two years as a starter for the Denver Broncos. He went 13-11 in his 24 starts for Denver, completing 59.3 percent of his passes for 5,686 yards, 30 touchdowns and 14 interceptions.
While those numbers are respectable, the Broncos opted to go in a different direction last year and traded Siemian to the Minnesota Vikings while signing veteran Case Keenum.
The 27-year-old Siemian did not take a single snap in 2018 as Kirk Cousins' backup.
Chris Tomasson of the St. Paul Pioneer Press' reported on Monday that Minnesota was looking to re-sign Siemian. However, the Vikings are now left with 2017 undrafted free agent Kyle Sloter as their backup.
New York needed to address its depth chart with 39-year-old journeyman Josh McCown's future up in the air.
It's not clear if the Jets would pursue McCown if he opted to give it another go, but the team now has added a veteran to the quarterback room.
Oddly enough, Siemian is not the only former 250th overall pick the Jets have signed in recent years, per Mehta:
Darnold started 13 games for the Jets as a rookie in 2018, missing three games due to a strained right foot.