Grant, who has never missed a game in his four-year NFL career, suffered an ankle injury in Washington's regular-season finale. As a free agent, Grant had agreed on a four-year, $29 million deal with the Ravens, only to have that deal voided last Thursday.
His agent told ESPN last Friday that Grant has been working out and running routes, and that his ankle was cleared by renowned surgeon Dr. Robert Anderson and is not an issue.
"If there was an issue with my ankle, I wouldn't have passed the physical with the Colts and the Raiders," Grant said. "I can't control what [the Ravens] did. I can't control their assessment of my ankle. I've gone to other places. I've seen multiple doctors and I've passed physicals so that should pretty much speak for itself."
Grant, 27, a fifth-round pick in 2014, caught 84 passes for 985 yards and four touchdowns during his four seasons with Washington.
He provides much-needed depth to a Colts receiving corps that is in need of help; Donte Moncrief signed with the Jacksonville Jaguars and Kamar Aiken is a free agent. T.Y. Hilton and Chester Rogers are the only receivers under contract who caught a pass for the Colts last season.
Cornerback Pierre Desir is returning to the Colts on a one-year deal, a source confirmed to ESPN on Tuesday. The Indianapolis Star first reported the deal.
Desir was limited to nine games in 2017 due to torn pectoral muscle. He was the team's No. 2 cornerback at the time of the injury.
Also on Tuesday, the Colts signed veteran center/guard Matt Slauson.
Slauson, 32, suffered a torn biceps in a Week 7 game against the Denver Broncos last year, forcing him to the injured reserve list and ending his season.
In his first season with the Chargers in 2016, Slauson solidified the center spot by starting all 16 games and playing 964 snaps there. The Chargers moved him to his more natural position of left guard last year, where he made seven starts and played 402 snaps on offense.
After a slow start in free agency, the Colts have added tight end Eric Ebron in addition to trading the No. 3 overall pick to the New York Jets for the No. 6 selection and two second-round picks in next month's draft, and another second-round pick in the 2019 draft.
The Colts also signed defensive lineman Denico Autry on March 15.
ESPN's Mike Wells, Eric D. Williams and Jamison Hensley contributed to this report.