The directive to identify new ways to hasten the demise of the Islamic State, also known as ISIS and ISIL, has been widely anticipated by military commanders, who have begun drafting classified options to increase the pressure on the militant group, especially in Raqqa and Mosul, the stronghold in Iraq. Work on the directive was described by several current and former officials who are close to the White House and who requested anonymity because they were not authorized to discuss the administration’s internal deliberations. The White House had no comment. The man charged with overseeing this re-examination of American defense is Mr. Mattis, a retired Marine Corps four-star general who commanded American forces in the Middle East and will be working in partnership with Gen. Joseph F. Dunford Jr., the chairman of the Joint Chiefs of Staff. The men have known each other for years and Mr. Mattis used to be General Dunford’s commanding officer while in the Marines. Mr. Mattis will face multiple challenges. As an emissary to longstanding allies in Asia and Europe, he has staked out a position as the Trump administration’s reassurer-in-chief. One of Mr. Mattis’s first moves as defense secretary was to phone the NATO secretary general to assure him that he strongly supported the alliance that Mr. Trump has criticized as “obsolete.” Mr. Mattis will fly to Asia next week on a trip to allay concerns in Japan and South Korea that the United States might abandon longstanding commitments to their security. A week after that, Mr. Mattis is expected to make another reassurance trip — this one to Europe — to meet with counterparts at NATO in Brussels and then at a security conference in Munich. Lawmakers and even some members of the military are hoping that Mr. Mattis can also serve as a counterweight on some of the new administration’s more hard-line positions. In a classified operations center at one Special Operations headquarters, a photo of Mr. Mattis is taped to a board with various captions written underneath. On Thursday morning, the caption read: “Watch over us.” During his first visit to the Pentagon, Mr. Trump will conduct a ceremonial swearing-in of Mr. Mattis and is expected to sign the new directives and have a short meeting with the Joint Chiefs of Staff. Mr. Trump and Mr. Mattis appear to have some positive chemistry. They were seen chatting warmly on the reviewing stand during the inaugural parade. The new commander-in-chief relishes referring to “Mad Dog” Mattis at every opportunity, even though the retired general does not like that nickname and insists it is no more than a media invention. And while they agree on the need for more military spending, some of the defense secretary’s views are at odds with his new boss, including his skepticism of Russia’s intentions, his traditional support for allies and flat opposition to the use of torture in interrogating terrorists. The day before Mr. Mattis came to work at the Defense Department, he issued a statement to the Pentagon work force that cast the United States as a bulwark of the international order and the guardian of important alliances. In contrast to the “America First” oratory emanating from the White House, Mr. Mattis vowed that the Pentagon would work “for an America that remains a steady beacon of hope for all mankind.” “General Mattis is prepared to give the president the best advice he can as secretary of defense even if it’s not something the president wants to hear,” said Senator Jack Reed of Rhode Island, the senior Democrat on the Armed Services Committee, who spoke to Mr. Mattis on Tuesday. “The question is, how long can he do that if he’s not being responded to.” Crafting a plan to step up the fight against the Islamic State is the most urgent task facing Mr. Mattis. When President Barack Obama left office, half of Mosul remained in the hands of the militants. Tens of thousands of American-backed Syrian Kurdish and Arab fighters were closing in on Raqqa, but there was no agreement on which force should seize the capital itself. The potential options include expanding the use of American Special Operations forces, raising the troop ceilings on United States forces in Iraq and Syria and having the White House delegate more authorities to the Pentagon and its commanders in the field, to speed up decision-making. A difficult decision also confronts the Pentagon on whether to risk alienating Turkey by arming the Syrian Kurds for the Raqqa battle, or whether to cobble together a more diverse force that could include Turkish troops, Turkish backed opposition groups and perhaps even elements of the Army’s 82nd Airborne Division, as well as Apache helicopters and artillery. Turkey considers the Syrian Kurds terrorists and has been trying to forge closer ties with the Trump administration. Expanding the American military will also pose challenges because of the soaring cost of some key weapons programs and the ambitious scope of the buildup Mr. Trump is seeking. As the steward of the Pentagon’s nearly $600 billion annual budget, Mr. Mattis will face tough choices, as it seems unlikely that the additional spending Mr. Trump plans for the armed forces can pay for all of the ambitious programs he has promised. During the campaign, Mr. Trump called for a Navy of 350 ships, up from the current fleet of 272, and to expand the Army to 540,000 troops, an increase of about 65,000. The Air Force and Marines would also grow. Funding such a military building would be costly. While the Pentagon has yet to outline its spending under the new administration, a paper by Senator John McCain, who heads the Armed Services Committee and is advocating a similar buildup, calls for spending $430 billion more than is currently planned, for the next five years. Other directives in the works could affect the military. Mr. Trump told ABC News on Wednesday that he would “absolutely do safe zones” in Syria for refugees fleeing the violence there. A draft executive order obtained by The New York Times calls for Mr. Mattis, along with Secretary of State Rex Tillerson, to produce a plan within 90 days for safe zones in Syria. In the past, American military officials have warned that such a move would escalate the American involvement in the war in Syria, something the Obama administration staunchly opposed. Military experts are looking to see who will join Mr. Mattis’s team at the Pentagon — and how many are picked by the new defense secretary himself. Mr. Mattis’s chief of staff will be Kevin M. Sweeney, a retired rear admiral whom Mr. Mattis has known for years. His senior military assistant will be Rear Adm. Craig S. Faller, who previously served as the top operations officer at Central Command when Mr. Mattis was in charge there.