President Biden and his top aides have engaged in an increasingly awkward dance in recent days, prodding Israel to change its tactics in the war in the Gaza Strip while still offering it robust public support.

Mr. Biden said last week that Israel was losing international support because of its “indiscriminate bombing” of Gaza, a much more critical assessment than his earlier public statements urging greater care to protect civilians. On Monday, Defense Secretary Lloyd J. Austin III, in his second visit to Israel since the Hamas-led Oct. 7 attacks, sought to take the temperature down a few degrees.

Meeting with Prime Minister Benjamin Netanyahu, Defense Minister Yoav Gallant and other top Israeli officials, Mr. Austin discussed in detail how Israeli forces will transition to the next phase of the war, a shift U.S. officials believe will lower the risk to civilians.

Mr. Austin is a retired four-star head of the Pentagon’s Central Command, overseeing U.S. military operations in the Middle East, and his word carries weight with Israel’s leaders, many of whom, like Mr. Gallant, are also former army generals.

Speaking to reporters after daylong meetings, Mr. Austin called U.S. support for Israel “unshakable,” and endorsed its campaign to destroy the ability of Hamas, which controls Gaza, to wage military operations in the difficult urban terrain. But he also repeated a message he has increasingly made of late: Israel will be left less secure if its combat operations turn more Palestinians into Hamas supporters.

“Israel has every right to defend itself,” he said, standing alongside Mr. Gallant. “As I’ve said, protecting the Palestinian civilians in Gaza is both a moral duty and a strategic imperative.”

Mr. Austin’s visit was part of a full-court press by the Biden administration to urge Israeli officials to wrap up the “high-intensity” phase of the war and begin carrying out more targeted, intelligence-driven missions to find and kill Hamas leaders, destroy the tunnels used by the militant group and rescue the people taken hostage on Oct. 7.

While U.S. officials have not publicly discussed a timetable, privately they say that Mr. Biden wants Israel to switch to more precise tactics in about three weeks.

Asked on Monday about the timeline of Israel’s campaign — a subject of intense discussions among U.S. officials in recent days — Mr. Austin demurred. “This is Israel’s operation, and I’m not here to dictate timelines or terms,” he said.

But Mr. Gallant indicated that Israeli officials were taking American concerns seriously: As the Israeli military achieves its objectives in different parts of Gaza, he said, it may be able to allow Gazans to begin returning to their homes. The vast majority of the population in Gaza have been forced from their homes.

Mr. Austin seemed to lean into that response, as if to bridge a divide, noting that every major military campaign has phases.

“The most difficult part is, as you shift from one phase to the next, making sure that you have everything accounted for and you get it right,” the defense secretary said. “That requires detailed planning and very thoughtful planning.”

Mr. Austin acknowledged how complicated the battle space is in southern Gaza for Israeli soldiers. As they have in northern Gaza, he said, Hamas fighters have used human shields as protection, and operated out of mosques, hospitals and schools.

But that is all the more reason, Mr. Austin told Israeli commanders in their closed meetings, that they must be as precise and disciplined as possible as they dismantle Hamas and its infrastructure, a senior Pentagon official said, speaking on the condition of anonymity to discuss internal talks.

Mr. Austin is deeply familiar with the painful lessons the American armed forces learned in the past two decades as they moved from major ground wars in Iraq and Afghanistan to more targeted operations, and he said he had shared those lessons with Israeli officials. “We also have some great thoughts about how to transition from high intensity operations to a lower intensity and more surgical operations,” he said.

Mr. Austin has had help walking the Israelis through the details. He was joined in his meetings on Monday by Gen. Charles Q. Brown Jr., the chairman of the Joint Chiefs of Staff. And the top U.S. commander in the Middle East, Gen. Michael E. Kurilla, met with senior Israeli officers last Friday, the Pentagon’s Central Command said.

American officials acknowledged that however encouraging Mr. Gallant’s suggestion that Israel was near the point of moving to a lower-intensity phase in northern Gaza, the road ahead was still very difficult.

As Mr. Austin walked to his motorcade Monday evening to fly on to his next stop on his Mideast trip, he shook hands again with Mr. Gallant, and offered a soldier-to-soldier salute: “Keep your head down, Mr. Minister.”