The Reds also reached deals Friday with right-handers Tanner Roark ($10 million), Anthony DeSclafani ($2,125,000) and Michael Lorenzen ($1.95 million), second baseman Scooter Gennett ($9,775,000) and shortstop Jose Peraza ($2,775,000).
Left-hander Alex Wood is Cincinnati's only arbitration-eligible player without a deal. He asked for a raise from $6 million to $9.65 million, and the team offered $8.7 million.
The Reds acquired Puig and Wood last month in a trade with the Los Angeles Dodgers. Cincinnati got Roark in a deal with Washington.
Puig hit .267 with 23 home runs and 63 RBI last season.