Per ESPN's Adrian Wojnarowski, the board of governors discussed delaying both events during a call with Silver on Friday.
Speaking to ESPN's Rachel Nichols (h/t Brian Windhorst of ESPN.com) on Thursday before the draft lottery, Silver said the previously discussed Dec. 1 start date for the 2020-21 season was "feeling a little bit early to me."
The commissioner explained the league's "No. 1 goal is to get fans back in our arenas," and moving back the start of next season could increase the chances of that happening.
One potential reason for moving the draft and free agency back stems from uncertainty related to the salary cap.
Marc Berman of the New York Post wrote in May that the cap was certain to decrease from its initial projection of $115 million for next season.
Wojnarowski reported Thursday that some front office executives are "privately concerned about the lack of salary-cap and luxury-tax projections" being in place before the draft.
The draft is scheduled to take place on Oct. 16, with free agency tentatively set to begin on Oct. 18 at 6 p.m. ET.
If free agency remains on that date, it could come five days after the season concludes if the NBA Finals goes to seven games.