January 25 update: Those in bold are the ones that press reports have said are currently being vetted.

First iteration (announced May 18, 2016):
1. Don Willett, Texas Supreme Court
2. Allison Eid, Colorado Supreme Court
3. Joan Larsen, Michigan Supreme Court
4. Thomas Lee, Utah Supreme Court
5. David Stras, Minnesota Supreme Court
6. Thomas Hardiman, 3rd Circuit U.S. Court of Appeals
7. Raymond Kethledge, 6th Circuit U.S. Court of Appeals
8. Diane Sykes, 7th Circuit U.S. Court of Appeals
9. Steven Colloton, 8th Circuit U.S. Court of Appeals
10. Raymond Gruender, 8th Circuit U.S. Court of Appeals
11. William Pryor, 11th Circuit U.S. Court of Appeals

Addendum (announced Sept. 23, 2016):
12. Edward Mansfield, Iowa Supreme Court
13. Keith Blackwell, Georgia Supreme Court
14. Charles Canady, Florida Supreme Court
15. Robert Young, Michigan Supreme Court
16. Federico Moreno, U.S. District Court for the Southern District of Florida
17. Amul Thapar, U.S. District Court for the Eastern District of Kentucky
18. Margaret Ryan, U.S. Court of Appeals for the Armed Forces
19. Timothy Tymkovich, 10th Circuit U.S. Court of Appeals
20. Neil Gorsuch, 10th Circuit U.S. Court of Appeals
21. Mike Lee, U.S. senator from Utah