Israeli troops and tanks pushed into parts of Gaza City, in the territory's north, and battled Palestinian freedom fighters who said they deployed explosives along with rocket and mortar fire against the Israeli forces.
The Islamist movement's political chief warned that developments on the ground could threaten talks, even as mediators Egypt and Qatar were due to host new meetings this week, according to officials.
Hamas's armed wing, the Ezzedine Al-Qassam Brigades, described the Gaza City combat as "the most intense in months".
On the other hand, Palestinians vacated Gaza City as the Israeli military expanded an evacuation order and mediators intensified efforts to end the nine-month war between Israel and Hamas fighters.
The United Nations said tens of thousands of people were affected by the evacuation orders.
Witnesses said messages on loudspeakers urged civilians to leave Gaza City's Al-Daraj and Al-Tuffah neighbourhoods.
Women carried the smallest children in their arms, while those old enough lugged backpacks or walked with bags slung over their shoulders through dusty streets already devastated by fighting in the war's earliest days.
The Israeli military called on Palestinians to leave parts of Gaza City's west, expanding its evacuation zone in the territory's biggest city with the third order in less than two weeks.
"Where do we go?" Abdullah Khammash asked in the street after fleeing Gaza City's Rimal neighbourhood. "They tell us to leave this area and go to another area, and then they come from that area."
The military ordered Gaza City's Al-Ahli Arab Hospital evacuated after "a large amount of firing from drones" nearby on Sunday, the Episcopal Church's Jerusalem Diocese said in a statement, adding that the facility was now "out of operation".
Only a minority of Gaza's hospitals are even partly functioning, the UN says, and the newly displaced face "critical levels" of needs such as safe drinking water.
The war started with Hamas's October 7 attack on southern Israel that resulted in the deaths of 1,195 people, mostly civilians, according to an AFP tally based on Israeli figures.
The militants seized 251 hostages, 116 of whom remain in Gaza, including 42 the military says are dead.
Israel responded with a military offensive that has killed at least 38,193 people in Gaza, also mostly civilians, according to data from the territory's health ministry.
The toll includes at least 40 deaths over the previous 24 hours, it said.
Diplomatic efforts to halt the fighting aim for an initial six-week ceasefire that would see some hostages in Gaza freed in exchange for Palestinian prisoners in Israeli jails, but talks would continue for a comprehensive deal to end the war.
Hamas has signalled it would drop its insistence on a "complete" ceasefire, a demand Israel has repeatedly rejected.
Israeli Prime Minister Benjamin Netanyahu's office reiterated in a statement that "any deal will allow Israel to return and fight until all the goals of the war are achieved".
- Evacuation order -
The military said Israeli forces were carrying out a "counterterrorism operation" against Hamas and Islamic Jihad militants in the area of the Gaza City headquarters of the United Nations agency for Palestinian refugees, UNRWA, which had no immediate comment.
In Gaza City's Shujaiya district, subject to evacuation orders since June 27 and where battles have raged for nearly two weeks, the military said it had "eliminated dozens" of militants including in air strikes.
Israel in early January said it had dismantled Hamas's "military framework" in northern Gaza, but militants have since regrouped -- underscoring the difficulty of destroying the group, which Netanyahu says is one of the goals.
A Hamas statement on Monday accused Netanyahu of trying to thwart an agreement.
While the Islamist movement "demonstrates flexibility... Netanyahu continues to place more obstacles in front of the negotiations, escalates his aggression and crimes against our people" to derail a deal, the statement said.
The Qatar-based political chief of Hamas, Ismail Haniyeh, said he had made "urgent contact" with mediators over events in Gaza which could "reset the negotiation process to square one", a Hamas statement said.
Israeli protesters have also accused Netanyahu of prolonging the war. They have stepped up their rallies, gathering in the tens of thousands to demand a hostage release deal and new elections.
Opposition leader Yair Lapid said Monday that a hostage deal "has to take place" and has support from "a large majority of the people".
Israel's President Isaac Herzog made a similar comment a day earlier.
Netanyahu's hard-right political allies have threatened to leave the government if he agrees to stop the fighting before Hamas is eliminated.
Egypt's state-linked Al-Qahera News said Cairo would host Israeli and US delegations in the latest effort to reach a truce.
A source with knowledge of the talks told AFP that US and Israeli intelligence chiefs would travel to Doha on Wednesday for discussions on a ceasefire deal, adding that they would meet with Qatari Prime Minister Mohammed bin Abdulrahman Al-Thani.
Alongside the Gaza war, Israel has exchanged near daily cross-border fire with Hamas's Lebanese ally, the Iran-backed Hezbollah.
Those exchanges have intensified, raising fears of a wider, all-out war.
On Monday, the Israeli military said an air strike had killed a Hezbollah operative in southern Lebanon. Hezbollah also announced the fighter's death, without elaborating.
Israel's military later reported that around 24 projectiles had been fired towards Israel from Lebanon.