The Rangers tried to acquire Upton in previous seasons but couldn't finalize a deal.
With the Rangers planning on moving left fielder Shin-Soo Choo to right field, they need another corner outfielder with power. Last season, Upton played in a career-high 154 games and hit .270 while also setting career highs in home runs (29) and RBIs (102).
Upton has a limited no-trade clause in his contract to four teams, but the Rangers are not one of them.
In addition to Upton's skills on the field, the Rangers are attracted to his contract. He has one year remaining on a six-year, $51.2 million contract extension he signed while playing with Arizona. Finding players with at least one year remaining on their contracts is appealing to the Rangers in this trade market.
The Braves are retooling their club after a 27-40 second-half collapse and already have traded outfielder Jason Heyward to the St. Louis Cardinals in a four-player deal that brought right-hander Shelby Miller to Atlanta.
The Rangers would also be interested in acquiring a pitcher from the Braves in a deal, a source said.
A source also said the Rangers placed a bid for Korean left-handed starter Yang Hyeon-Jong, and MK Sports reported the Rangers placed the highest bid at $1.5 million. A Rangers official said the team won't know if it won the Tigers accept the bid until this week.
In eight seasons pitching for the Kia Tigers, Hyeon-Jong went 62-42 with a 4.33 ERA.