The contract comes with a player option for 2015 and should Chapman opt out, he would be arbitration-eligible rather than a free agent.
The deal was first reported by Yahoo! Sports.
Chapman traveled to Cincinnati on Sunday and sources say the deal could be completed as early as tonight and announced tomorrow.
Chapman defected from Cuba on July 3 when he walked away from a hotel in Amsterdam on the eve of an international tournament in Rotterdam.
The Reds were a quiet but consistent contender for the 22-year-old, whose 102-mph fastball drew significant interest from the likes of the Boston Red Sox, Florida Marlins, Los Angeles Angels, Oakland Athletics, Toronto Blue Jays and Washington Nationals.
Chapman captured scouts imagination at the 2009 World Baseball Classic, where he hit triple-digits on the radar gun four times. During a four-year stint in Cuba's National Series, he had a 24-21 record with a 3.72 ERA. Almost as notable as his 379 strikeouts in 341 2/3 innings are his 210 walks.
The 6-foot-4, 205-pound pitcher allayed some of the concerns about his control at a Dec. 15 workout in Houston, where he threw roughly 45 pitches with markedly better control, according to one MLB scout who was present.
A source familiar with the Reds says Cincinnati was not overly concerned with Chapman's control issues -- he registered 5.37 walks per nine innings in Cuba -- and said that the Reds likened him to a young Randy Johnson, who also needed time to harness his left-handed power before progressing to a Cy Young Award-winning career. Though some scouting reports project Chapman as a reliever, the Reds are said to consider him a starter, like Johnson.
Some teams expressed concerns about Chapman's maturity and makeup, but sources say the Reds felt comfortable with Chapman.
He will most likely need some preparation time in the minor leagues before joining an already deep and young rotation that includes Edinson Volquez and Johnny Cueto, but, as one veteran international scout told SI.com last fall, "He's a left-handed pitcher who throws 100 miles per hour. You can travel the world and not find that."
Now, sources say, one will be one display in Ohio.