Smith said Thursday he anticipates a meeting to happen "as soon as possible," NFL Media's Omar Ruiz reported, but no meeting currently is set.
Last week, a judge sentenced Smith to 12 days in county jail with a one-day credit, according to a court document obtained by NFL Media. However, that sentence can be served on work crew on Mondays, starting July 28.
Goodell is expected to make a decision on a potential suspension after meeting with the 49ers linebacker.
The Niners star took a leave of absence from the team last season and voluntarily checked into a treatment facility following the arrest for DUI. He missed five games in total.
Smith said he's been sober since he went to rehab in September.
"I've been able to maintain (sobriety). It's going good," Smith told reporters.
The stud outside linebacker said he was in the "best shape" of his career entering camp. The question now is how much of the season he'll spend watching his teammates play.