Former San Francisco 49ers star Kaepernick has been out of the league since the end of the 2016 season, a campaign in which he attracted controversy by kneeling during the United States national anthem in protest against racial injustice and police brutality.
He filed a grievance against the NFL in 2017, accusing owners of colluding to keep him out of a job. Kaepernick settled that grievance in February.
After repeatedly claiming he was ready for a return, the 32-year-old was granted a workout for NFL teams in Atlanta on Saturday - although the location was changed as his representatives criticized "recent decisions made by the NFL".
Brady's New England Patriots were among the teams interested as the workout went ahead, with Kaepernick addressing the media afterwards.
"We'll be waiting to hear from [NFL commissioner] Roger Goodell, the NFL, the 32 teams," he said. "The ball's in their court, we're ready to go."
Veteran superstar Brady hopes the event will see Kaepernick granted a route back into the league.
"It's great that he got an opportunity. Hopefully, he makes the most of it," Brady told The Greg Hill Show on WEEI on Monday.
"It's like everyone in the NFL. It's a great privilege to play and do something you love to do. I've felt that way for a long time.
"Hopefully, he gets an opportunity. That's how I feel. I like Colin. I got to know Colin a little bit. I know he's obviously hoping he can play."
The Patriots beat the Philadelphia Eagles 17-10 on Sunday and Brady said the offense could "do everything better".
Expanding on his comments, the 42-year-old claimed New England are now a team dependent on their defense, despite his status as one of the greatest quarterbacks of all time.
"The reality is it's a team sport. It's complementary football," he said. "The strength of our team is our defense and our special teams.
"So on offense, we've just got to take advantage of when we get opportunities and understand where our strengths lie and try to play to them."