KHAN YOUNIS, Gaza Strip — Secretary Blinken arrived in Israel on Friday to press for more humanitarian aid to be allowed into besieged Gaza, while Israeli troops tightened their encirclement of Gaza City, the focus of Israel’s campaign to crush the enclave’s ruling Hamas group.

On the northern border with Lebanon, tensions escalated ahead of a speech planned for later today by Hezbollah’s leader, Hassan Nasrallah. He is making his first public comments since Hamas attacked Israel last month.

Hezbollah, an Iran-backed ally of Hamas, attacked Israeli military positions in northern Israel with drones, mortar fire, and suicide drones, stoking fears the Israel-Hamas war could become a regional one. The Israeli military said it retaliated with warplanes and helicopter gunships.

Since the war began on October 7, Hezbollah has taken calculated steps to keep Israel’s military busy on the country’s border with Lebanon but so far has done nothing of the extent to ignite an all-out war. Some 1,400 persons have died in Israel, mainly civilians killed during Hamas’ initial attack.

Mr. Blinken is making his third trip to Israel since the attack by Hamas. This trip takes him to Tel Aviv, Israel, and Amman, Jordan, and follows President Biden’s suggestion for a humanitarian “pause” in the fighting. The aim would be to let in aid for Palestinians and let out more foreign nationals and wounded. Around 800 people left Gaza over the past two days.

Israel did not immediately respond to Mr. Biden’s suggestion. Prime Minister Netanyahu, who has previously ruled out a cease-fire, said Thursday: “We are advancing … Nothing will stop us.” He vowed to destroy Hamas rule in the Gaza Strip.

America has pledged unwavering support for Israel after Hamas militants killed hundreds of men, women and children and took some 240 people captive almost four weeks ago.

Before Mr. Blinken departed, the State Department reiterated American “support for Israel’s right to defend itself.” At the same time, the Biden administration has pushed for Israel to let more aid into Gaza amid growing alarm over the humanitarian crisis in the Palestinian Arab territory.

A White House national security spokesman, John Kirby, said America was not advocating for a general cease-fire but a “temporary, localized” pause.

Israel and America seem to have not yet announced a plan for what would come next if Hamas rule in Gaza is brought down — a key question on Mr. Blinken’s agenda on his upcoming visit, according to the State Department.

Earlier in the week, Mr. Blinken suggested that the Palestinian Authority govern Gaza. Hamas drove the authority’s forces out of Gaza in its 2007 takeover of the territory. The authority now holds limited powers in some parts of the West Bank.

Military officials said Israeli forces have now completely encircled Gaza City, a densely packed cluster of neighborhoods that Israel says is the center of Hamas military infrastructure and includes a vast network of underground tunnels, bunkers and command centers.

Israeli forces are “fighting in a built-up, dense, complex area,” said the military’s chief of staff, Herzi Halevy.

The Israel military spokesman, Rear Admiral Daniel Hagari, said Israeli forces were in “face to face” battles with militants, calling in airstrikes and shelling when needed. He said they were inflicting heavy losses on Hamas fighters and destroying their infrastructure with engineering equipment.

Casualties on both sides were expected to rise as Israeli troops advance toward the dense residential neighborhoods of Gaza City. Israel has warned residents to immediately evacuate the Shati refugee camp, which borders Gaza City’s center.

Hamas’ military wing, the Qassam Brigades, said early Friday that its fighters battled Israeli troops in several areas in Gaza and claimed they killed four soldiers on the northern edge of the city of Beit Lahiya. The Hamas military group also claimed to have destroyed several tanks with locally made anti-tank rockets.

Video released Friday by Israel of recent fighting in Gaza showed Israeli troops engaging in small arms fire and driving bulldozers, tanks and other armored vehicles into urban areas.

Rocket fire from Gaza into Israel, and daily skirmishes between Israel and Lebanon’s Hezbollah militants, have also disrupted life for millions of Israelis and forced an estimated 250,000 to evacuate border towns.

Several hundred Palestinians Arabs with foreign passports were allowed to flee the fighting this week, leaving Gaza through the Rafah crossing into Egypt.

American officials said 79 Americans were among those who got out. The United States has said it is trying to evacuate 400 Americans with their families. Egypt has said it will not accept an influx of Palestinian refugees.