Nelson, taken No. 6 overall by the Colts, was among nine draft picks signed by Indianapolis, the team announced Friday.
Indianapolis also signed defensive ends Kemoko Turay and Tyquan Lewis, running back Nyheim Hines, wide receiver Daurice Fountain, running back Jordan Wilkins, wide receiver Deon Cain and linebackers Matthew Adams and Zaire Franklin.
Nelson was a unanimous All-American at Notre Dame and started 36 of 37 games in his career. Colts general manager Chris Ballard called the Outland Trophy finalist "the easiest pick I've ever been a part of."
"God made him a little bit different than He made everybody else," Ballard told the Indianapolis Star last week.
Indianapolis also locked up two of their second-round selections in defensive ends Turay and Lewis.
The 6-5, 253-pound Turay registered 20.5 tackles for loss, 15.5 sacks, four passes defensed, one forced fumble, three fumble recoveries and three blocked kicks while starting 10 of 44 games at Rutgers.
Lewis started 36 of 53 games at Ohio State. The 6-3, 269-pounder notched 36.5 tackles for loss, 23.5 sacks, five passes defensed and five forced fumbles.
Fourth-round pick Hines amassed 4,638 all-purpose yards at North Carolina State. He rushed for 1,112 yards and 12 touchdowns and added 26 receptions for 152 yards in 2017.
Fountain, a fifth-round selection out of Northern Iowa, started all 13 games last season and finished with 66 receptions for 943 yards and 12 touchdowns.
Wilkins, another fifth-round pick, started 14 of 38 games at Mississippi and racked up 1,751 yards and 14 touchdowns for his career. He also had 32 catches.
Cain, a sixth-round choice and a member of Clemson's national championship team in 2016, posted career numbers of 130 receptions for 2,040 yards and 20 touchdowns.
Adams and Franklin were each seventh-round selections. Adams had 7.5 sacks while starting 26 of 48 games at Houston, while Franklin had 8.5 sacks and 31.5 tackles for loss in four seasons at Syracuse.