The defensive end, 26, emerged through the International Player Pathway programme but was cut by the New Orleans Saints before the start of the 2018 season.
He has been signed by the Giants in the wake of their rookie minicamp the past weekend, and will now fight to make the final 52-man roster before the start of the season in August.
Jenkins, from Bath, is one of a number of recent British graduates of the International Player Pathway programme, which offers players from outside the U.S. a route into the league.
A former schoolboy rugby and soccer player, he is emulating other British sports stars in making the leap and trying to break through.
Former England international Christian Wade joined the Buffalo Bills in April just seven months after ending his career in rugby.
The ex-England sevens rugby captain Alex Gray has a reserves future contract with the Atlanta Falcons and his former teammate Christian Scotland-Williamson joined the Pittsburgh Steelers through the programme in 2018.
Their compatriot Jay Ajayi, from London, won the Superbowl in 2018 with the Philadelphia Eagles and is now a free agent.