Carolina will have the option of deciding which of its first-round selections to give to the Rangers as part of the deal prior to the draft.
Skjei has accumulated 23 points, 41 penalty minutes, and 91 blocked shots across 60 games this season.
The 25-year-old, along with Vatanen, are joining a Hurricanes defense corps that includes Jaccob Slavin, Jake Gardiner, Trevor van Riemsdyk, and Haydn Fleury. Dougie Hamilton and Brett Pesce are both out with long-term injuries.
Carolina currently holds the second Eastern Conference wild-card spot but is tied with the Columbus Blue Jackets at 74 points apiece. The Rangers and Florida Panthers are also not far behind.