Along with Osweiler, the Colts have also worked out quarterbacks Matt Cassel and Brandon Weeden this week, according to ESPN and reports.
The Colts are looking for a veteran backup quarterback to play behind starter Jacoby Brissett following the surprising retirement of Andrew Luck last weekend.
The only other quarterbacks on the Colts' roster are Phillip Walker and Chad Kelly. Walker has never played a snap in a regular-season game and Kelly is suspended for the first two games of the season for violating the league's personal conduct policy.
Colts coach Frank Reich gave the indication earlier this week that they would look to upgrade the backup spot.
"Those are all really good things to consider -- that we are considering very strongly," Reich said Monday. "Phil [Walker] has been here for a while so we have a lot of belief and confidence in Phillip. He knows our offense. He knows this locker room. Those are going to be the decisions that we have to make in the next week to 10 days."
The Colts will continue to do their due diligence in finding a backup quarterback, including possibly waiting until after roster cutdown to sign one.
Osweiler, 28, went 2-3 as a fill-in starter for the Miami Dolphins last season when Ryan Tannehill was injured. He gave the fan base hope in a period dubbed "Brocktober," which began with him throwing for a career-high 380 yards in an upset overtime victory against the Chicago Bears, but his play leveled off in the following games.
What Osweiler did do, however, is re-establish his career as a disappointing starter to become a reliable backup. He had signed a one-year, $880,000 contract with Miami and competed to make the roster as a backup. Despite a rough training camp, the Dolphins kept Osweiler as their No. 2 QB and he kept them afloat when Tannehill got hurt early in the season.
In 2018, Osweiler completed 63.5% of his passes for 1,247 yards, six touchdowns and four interceptions.
Osweiler's biggest NFL contribution came in a similar role. He went 5-2 as a starter in place of an injured Peyton Manning for the 2015 Denver Broncos, helping them earn the AFC's top seed en route to a victory in Super Bowl 50.
That season led to a $72 million contract with the Houston Texans, who traded him to the Cleveland Browns after one difficult season as the starter. The Browns released Osweiler just before the 2017 season, when he was again picked up by the Broncos and started five games.
Osweiler, who is entering his seventh season, was drafted by the Broncos in the second round (57th overall) of the 2012 draft.
ESPN's Cameron Wolfe contributed to this report.