Los Angeles will send two prospects -- neither of whom is prized 19-year-old left-hander Julio Urias -- to Cincinnati, the source told Bowden. The deal, which is pending Chapman's passing a medical examination, was first reported Monday by Fox Sports.
Reds president of baseball operations Walt Jocketty said last month that he hoped to move the hard-throwing left-hander before this week's winter meetings in Nashville, Tennessee.
"We go through this at the trade deadline at the end of July, and now we have a bunch of players with names out there, so it's uncomfortable enough," Reds manager Bryan Price said. "So if anything happens with any of our guys, it would be nice if it happened sooner rather than later, but that's just the nature of the beast."
The Reds finished last in the NL Central the past season with a 64-98 record.
The Dodgers saw right-hander Zack Greinke agree to a reported six-year, $206 million deal with the Diamondbacks on Friday. Greinke led the majors in ERA the past season while helping the Dodgers win their third straight division title. He opted out of his contract to become a free agent.
Los Angeles already has an established closer, with Kenley Jansen recording 36 saves in 2015 and 44 in 2014.
Presuming the Dodgers plan to keep both relievers, new manager Dave Roberts could have a difficult time getting one of them to accept a setup role. Both pitchers, who will be free agents after the season, are eligible for arbitration and figure to make more than $10 million in 2016.
In six years in the majors, Chapman has a 2.17 ERA with 146 saves. He has averaged 15.4 strikeouts per nine innings.
Chapman threw the 62 fastest pitches in the big leagues the past season, according to Major League Baseball's new Statcast computer system. His fastest pitches ranged from 102.36 mph to 103.92 mph.
ESPN Staff Writer Mark Saxon and The Associated Press contributed to this report.