Terms were not disclosed, but the deal is worth $8.25 million, according to multiple reports.
Nash, 29, had a career-high 41 points (15 goals, 26 assists) last season with the Boston Bruins. He'll count $2.75 million against the salary cap in Columbus.
He was a plus-16 on the season after entering the year as a career minus-19.
Nash only had one assist in nine playoff games, however, as Boston fell in the second round to the Tampa Bay Lightning.
The Associated Press contributed to this report.