The two-time All-Star came to the Reds last July in a trade with Kansas City. He filled in as the closer when Aroldis Chapman developed a tired shoulder and had four saves in six chances overall with a 2.82 ERA.
Now the Reds have the option of turning Chapman into a starter, which was the plan last season until Madson blew out his elbow. The team has told Chapman to prepare for next season as a starter, although it hasn't committed to Broxton as the closer.
"Nothing's in stone right now," assistant general manager Bob Miller said. "When we talked to Jonathan we said he was going to be at the back end of the bullpen. What happens depends on spring training and how things play out in the offseason."
Broxton wanted a multiyear deal so he could settle in one place. He didn't insist on assurances he'd be a closer.
"I went into the offseason with an open mind," Broxton said. "I've got experience in both roles. Even if Chapman doesn't work out as a starter, he can come back in and fill in as the closer. I'll be happy to throw the eighth (inning). It doesn't matter. You saw what he did last year."
The 28-year-old Upton hit .246 with 78 RBIs for the Rays last season. He will replace free agent Michael Bourn as the Braves' center fielder and should provide much-needed power from the right side.
Upton's first full season with Tampa Bay was 2007, when he hit a career-best .300 with 24 homers and 22 stolen bases. His 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.
Elsewhere, the Boston Red Sox made a flurry of minor moves, trading right-handers Zach Stewart and Sandy Rosario as well as third baseman Danny Valencia.
Stewart was sent to Pittsburgh and Rosario to Oakland for players to be named. Valencia was shipped to Baltimore for cash.
Boston also hired Greg Colbrunn as hitting coach.
Athletics reliever Pat Neshek agreed to a one-year contract that avoided salary arbitration, while Kansas City traded right-hander Vin Mazzaro and first baseman Clint Robinson to the Pirates for minor league pitchers Luis Santos and Luis Rico.
The Chicago Cubs designated right-hander Casey Coleman for assignment to make roster room for newcomer Scott Feldman, who agreed to a $6 million, one-year contract the previous day.
Feldman's former team, the Texas Rangers, acquired right-hander Cory Burns from San Diego for a player to be named or cash.
AP Sports Writers Charles Odum, Greg Beacham and Joe Kay contributed to this report.