Dunford, a former commandant of the Marine Corps and commander of coalition troops in Afghanistan, is expected to serve out his term as Joint Chiefs chairman, which ends October 1.
Dunford's spokesman, Col. Patrick Ryder, said all indications are that Dunford will serve his full term.
President Donald Trump said he'll be making an announcement on Saturday concerning the Joint Chiefs of Staff. He later commanded the 10th Mountain Division at Fort Drum in NY, the III Corps at Fort Hood in Texas, and U.S. Army Forces Command at Fort Bragg in North Carolina before taking the reins as chief of staff in August 2015.
Milley who holds degrees from Princeton University and Columbia University, is known to be liked by Trump. Trump said the date of transition is "to be determined".
Trump announced his picks for a new attorney general and a new United Nations ambassador on Friday and initially suggested he might make the military announcement at Saturday's annual Army-Navy football game in Philadelphia.
Dunford is a Marine Corps general. He has served in numerous command and leadership position, including in Iraq and Afghanistan.
Milley served as a Green Beret and commanded troops in Afghanistan. Milley made the decision to charge Bergdahl in 2015 with desertion and misbehavior before the enemy, for which Bergdahl ultimately received a dishonorable discharge but avoided prison time. Milley started the first U.S. Army-Navy ice hockey game in 2016 in advance of the 2016 Army-Navy football game. An infantry officer by training, he also commanded Special Forces units in a career that included deployments in the invasion of Panama in 1989; the multinational mission in Bosnia-Herzegovina to implement the Dayton Peace Accords; and the Iraq war. "And I look forward to that", Trump told reporters later Saturday, praising Milley after announcing his nomination.
Inside the Pentagon, Milley was seen as one of two top choices to succeed Dunford alongside the Air Force's top officer, Gen. David Goldfein, a veteran fighter pilot who has served as chief of staff of his service since 2016.