Chapman became available after the Reds' deal with the Los Angeles Dodgers fell through during baseball's winter meetings three weeks ago when it was learned Florida police investigated an accusation of domestic violence involving the Cuban left-hander.
Major League Baseball is currently investigating and Chapman could face suspension under the league's new domestic violence policy.
''We felt this was an opportunity to add a big arm to our bullpen, even though there are some things that are unresolved,'' Yankees general manager Brian Cashman said in a conference call. ''And we will respect that process as it plays out.''
Cashman said the Yankees have had interest in Chapman for several years, but a deal only became affordable when the price for the reliever dropped after the report was made public.
New York was able to protect its top minor leaguers in the deal, sending right-handers Caleb Cotham and Rookie Davis and infielders Eric Jagielo and Tony Renda to Cincinnati. Jagielo was New York's No. 1 pick in 2013 (26th overall).
The Yankees will have to wait to set up their enviable bullpen of Chapman, Andrew Miller and Dellin Betances until the investigation is done.
MLB has indicated it will take as long as it needs to thoroughly vet the matter. The league is also investigating domestic violence incidents involving Colorado's Jose Reyes and the Dodgers' Yasiel Puig.
Davie, Florida, police said there was ''insufficient evidence'' to charge Chapman in the disturbance between the pitcher and his girlfriend at his South Florida home. The Davie Police Department report listed the Oct. 30 matter as closed.
But after the report's release earlier in the month, state prosecutors said they were examining the possibility of criminal charges.
''Certainly there are some serious issues here that are in play,'' Cashman said. ''I acknowledge that's an area clearly of concern and I think it certainly is reflective of some of the acquisition price and there's risk, and I understand that.''
Chapman gives the Yankees the top three relievers by strikeouts and strikeouts per nine innings in the majors. Miller had 36 saves in his first year with New York and Betances made his second straight All-Star team as the setup man.
There was talk earlier in the offseason that the Yankees were looking to possibly move Miller for a starting pitcher, but Cashman is intent on keeping all three relievers as a ''real force at the back of our bullpen.''
''I'm sure we'll get a lot of interesting calls between now and whenever,'' Cashman said of potential trade offers.
Miller is entering the second year of a four-year, $36 million deal.
A 27-year-old left-hander, Chapman is eligible for free agency after next year's World Series. But that could be pushed back until after the 2017 season if he is suspended by MLB without pay for more than 45 days, giving the Yankees an extra year of control over one of baseball's best relievers.
After defecting from Cuba in 2009, Chapman spent the past six seasons with the Reds and saved 146 games in 164 chances. He had a 1.63 ERA this year, when he struck out 116 in 66 1-3 innings. Betances struck out a league-best 131 and Miller fanned 100.
Chapman threw the 62 fastest pitches in the big leagues this year, ranging from 103.92 mph to 102.36 mph, according to MLB's Statcast computer system.
He was an All-Star in each of the past four seasons, and Chapman, who is arbitration eligible, is certain to earn a big payday this offseason.
With the Reds undergoing a major retooling after a 98-loss season, Chapman joined Cincinnati stars Johnny Cueto, Todd Frazier and Mike Leake who have all been dealt since July. The club also appears to be looking to move Brandon Phillips.