Once finalized, Upton's deal will be worth $75.25 million over five years, Heyman reports. That would be the largest free-agent contract in team history.
Upton, 28, hit .246 with 28 homers, 78 RBIs and 31 stolen bases for Tampa Bay last season. He will replace free agent Michael Bourn as the Braves' center fielder and provide a needed right-handed power bat for the lineup.
Upton made his debut with Tampa Bay in 2004. His first full season was 2007, when he hit a career-best .300 with 24 homers and 22 stolen bases.
The speedy outfielder's home run totals have increased in each of the last three seasons, but he has hit below .250 with more than 150 strikeouts in four straight years.
Bourn was the Braves' leadoff hitter, but Upton is not expected to fill that role. The Braves might shop for a left fielder to hit leadoff, with Martin Prado expected to move from the outfield to third base following Chipper Jones' retirement.
Young shortstop Andrelton Simmons, who hit .289 with a .335 on-base percentage as a rookie, could be an option to bat leadoff. Prado was the normal No. 2 hitter.
The deal with Upton comes only five days before the start of baseball's winter meetings in Nashville, Tenn.
Upton is an expensive addition, but the big move is not a surprise.
General manager Frank Wren said after the season he would have funds available to make a notable offseason acquisition. Jones had a $14 million base salary last season and Bourn's was $6,845,000.
Wren said he entered the offseason with "a pretty narrow focus" because he only had to replace Bourn and Jones.
"Beyond that, our club is pretty solid," Wren said after the Braves' wild-card playoff loss to St. Louis.
Upton adds to what Wren said is a "dynamic young core" of players. He will join first baseman Freddie Freeman, right fielder Jason Heyward, catcher Brian McCann and second baseman Dan Uggla as power hitters.
McCann's status for the start of the season is uncertain. He had surgery on his right shoulder in October. The rehabilitation could carry into the first month of the season.
Gerald Laird signed a two-year, $3 million deal with Atlanta earlier this month to serve as McCann's backup. Laird, who hit .282 with AL champion Detroit, could open the season as the starter if McCann is not ready.
Upton's agent, Larry Reynolds, did not immediately return messages seeking comment.