The deal is for five years and worth $50 million, according to multiple reports.
Vaitai, who started 20 games over his four seasons with the Eagles, could be the Lions' replacement at right tackle for Rick Wagner, who was released Friday.
The 26-year-old Vaitai stepped in at left tackle for an injured Jason Peters midway through the 2017 season and held his own, including in the playoffs as the Eagles made their Super Bowl charge.
Vaitai, who has experience playing left and right tackle, has started four games in the two seasons since that title run with mixed success, but he has provided valuable depth since being selected in the fifth round out of TCU in 2016.
ESPN's Tim McManus contributed to this report.