At the top of the list is retired four-star Army General Jack Keane, Administration and congressional officials said. Keane retired in 2003 and served as vice chief of staff of the Army.
Unlike Mattis, Keane would not need a congressional waiver to serve as defence secretary because he's been out of military service more than seven years. He is widely respected in Congress, inside the Administration and by the President himself.
"He has strong relationships across the Administration and the President likes his TV profile," the senior White House official said. "He comes across as strong and competent - like a more partisan Mattis."
Having another retired general run the Defence Department could raise concerns about deterioration of civilian control over the military, as it did when Trump chose Mattis.
But multiple congressional officials told me that a Keane nomination would be met with broad support in Congress, given the even greater concerns on Capitol Hill about Trump's handling of national security amid a range of complex and growing threats.
"Trump needs somebody he respects and he respects General Keane," a senior GOP congressional aide told me. "I think he has the right skill set and attributes to be effective with this Administration. And he'll work well with the rest of the Cabinet."
Other names under discussion inside the White House include Senator Tom Cotton, Senator Lindsey Graham, former Treasury Department official David McCormick; and former Senator Jim Talent.
Depending on the party balance in the Senate after the November elections, Republican leadership may not want to risk losing one its caucus members, especially Cotton, who defeated a Democrat for his seat in 2014.
Graham has said repeatedly he is not interested in serving in the Cabinet. McCormick was considered during the transition for deputy positions in both the Defence and Treasury departments. Talent was also considered for defence secretary in late 2016.
Keane declined to comment. Sources said Keane has not had any recent discussions with White House officials about being considered for the job. Keane spoke broadly about his vision for national security and defence policy at today's conference at the Institute for the Study of War, a think-tank whose board he chairs.
Keane endorsed Trump's National Security Strategy and Mattis's own National Defence Strategy, released this year, which calls for a return to great power competition with "revisionist powers" Russia and China. Keane warned about Iran's regional expansion, noted North Korea's pledge to denuclearise and highlighted the continuing threat from radical Islamic groups, including Isis and al-Qaeda.
Tracking closely with Trump's own philosophy, Keane called for a reinvestment in the US military, which he said has suffered from years of insufficient funding and support.
