Israel attacked the ancient port city of Tyre in Lebanon on Wednesday after issuing its broadest evacuation order there so far, pressing on with its bombing campaign against Hezbollah even as Secretary of State Antony J. Blinken toured the region in pursuit of a diplomatic solution to the escalating conflict.

The airstrikes in Tyre, which wounded at least 16 people, according to Lebanon’s health ministry, came hours after Israel’s military warned civilians in a large portion of the southern city to move about 25 miles north. The evacuation area covered a densely populated stretch of the city, which was until recently a hub for people fleeing other parts of southern Lebanon. The extent of the damage was not immediately clear.

Mr. Blinken, who was in Israel on Wednesday morning before traveling to Saudi Arabia, got a personal but limited view of the deadly conflicts wracking the region when the launch of Hezbollah launched missiles forced him and other guests at the David Kempinski hotel in Tel Aviv into shelter rooms.

Just before leaving Tel Aviv on Wednesday, the secretary of state seemed to signal that the Biden administration was open to the possibility of approaches to stop the fighting between Israel and Hamas in the Gaza Strip, which American officials see as one of the keys to an overall de-escalation in the region.

In a departure from his usual talking points on Gaza cease-fire proposals, Mr. Blinken said that the United States was “looking at new frameworks of formulations and possibility.” He did not provide details, but a senior U.S. official said he was referring to the possibility that Israel might be willing to pause its Gaza offensive briefly in return for the return of a small number of hostages.