The Tampa Bay Lightning, who lost 2-1 to the Pittsburgh Penguins in Game 7 of the Eastern Conference Final on Thursday, will pick 27th.
The St. Louis Blues, who were eliminated from the Stanley Cup Playoffs with a 5-2 loss to the San Jose Sharks in Game 6 of the Western Conference Final on Wednesday, will pick 28th.
The draft will be held June 24 and 25 at First Niagara Center in Buffalo.
Five teams have multiple picks in the first round.
The Winnipeg Jets, who won the second pick in the lottery, also have the Chicago Blackhawks' pick at No. 22, acquired in the Feb. 25 trade for forward Andrew Ladd.
The Arizona Coyotes have the seventh pick and the New York Rangers' selection at No. 20, acquired in the March 1, 2015, trade for defenseman Keith Yandle.
The Carolina Hurricanes have their pick at No. 13 and the Los Angeles Kings' selection at No. 21, which they got in the Feb. 25, 2015, trade for defenseman Andrej Sekera.
The Toronto Maple Leafs, who won the lottery and will pick No. 1, also have the Penguins' pick from the Phil Kessel trade on July 1.
The Boston Bruins have the No. 14 pick and the Sharks' first-round selection, acquired in the June 30 trade for goaltender Martin Jones.
The order of the final two picks will be determined by the result of the Stanley Cup Final between the Penguins and Sharks.
1. Toronto Maple Leafs
2. Winnipeg Jets
3. Columbus Blue Jackets
4. Edmonton Oilers
5. Vancouver Canucks
6. Calgary Flames
7. Arizona Coyotes
8. Buffalo Sabres
9. Montreal Canadiens
10. Colorado Avalanche
11. New Jersey Devils
12. Ottawa Senators
13. Carolina Hurricanes
14. Boston Bruins
15. Minnesota Wild
16. Detroit Red Wings
17. Nashville Predators
18. Philadelphia Flyers
19. New York Islanders
20. Arizona Coyotes (from New York Rangers)
21. Carolina Hurricanes (from Los Angeles Kings)
22. Winnipeg Jets (from Chicago Blackhawks)
23. Florida Panthers
24. Anaheim Ducks
25. Dallas Stars
26. Washington Capitals
27. Tampa Bay
28. St. Louis
\29. Boston (from San Jose)
\30. Toronto (from Pittsburgh)
\- to be determined at end of Stanley Cup Final*