Smart will make $77million in the deal, which begins with the 2022-23 NBA season, according to reports by ESPN and The Athletic.
Smart posted a video of highlights from his Celtics career on his Instagram page Monday with the comment "Let's go 4 more..."
The 27-year-old has spent his entire seven-year NBA career with the Celtics after they selected him with the sixth overall pick in the 2014 draft.
He averaged a career-best 13.1 points and 5.7 assists per game last season.
Smart is set to make $14.3m next season, the last in his existing deal.