Green revealed in a podcast on Tuesday that he had been dealing with a torn groin muscle.
The players had a week to report to their teams and pass physicals.
Leonard last played on Jan. 13. He sat out the rest of the season to manage a quad injury and ended up playing only nine games for San Antonio.
Spurs doctors had cleared Leonard to return to play, but he sought second opinions and did not return, making his health a question mark following the trade.
Meanwhile, sources told ESPN's Chris Haynes that Leonard will not attend Team USA minicamp in Las Vegas next week. Spurs coach Gregg Popovich is the coach of that team.
Before the Toronto deal, sources had said that Leonard was seriously considering participating as a means to showcase the status of his recovery to prospective trade partners.