Many displaced Gazans are living "in open spaces, in parks", Juliet Toma of UNRWA, the UN relief agency for Palestinian refugees, told the BBC.
According to the United Nations Humanitarian Office (OCHA), at least 100,000 people have fled to Rafah, south of Gaza along the border with Egypt, in recent days.
UNRWA's Toma said the UN was being authorized to bring "limited assistance". But Gaza's humanitarian needs have grown enormously.
Israel says it is not limiting aid.
Toma said UNRWA "faces restrictions on access to areas of the Gaza Strip that we should be able to access".
On October 7, Israel launched its offensive in Gaza following the Hamas attack.
Humanitarian access to the 41 km (25 miles) long and 10 km wide enclave has been tightly controlled since the war began.
At the start of its military campaign, Israeli forces focused most of the fighting in northern Gaza. But recently, Israel has also launched an attack on Khan Yunis in southern Gaza which they see as a stronghold of Hamas.
According to the UN agency OCHA, the most recent outbreak of displacement in the town of Rafah was intense fighting in Khan Yunis and other parts of the territory.
Speaking from Rafah, Tom White, UNRWA's Gaza director, told the BBC there were "more than a million people" seeking safety in the city.
White said "hundreds of thousands of people" have "left no trace" of shelters for safety in Rafah - including UNRWA schools and other municipal facilities.
He said following your comments, hundreds of people are now sleeping in the open under small pieces of plastic due to overcrowding.
In their latest report, UNRWA said 1.9 million people have been displaced across the Gaza Strip since the war began.
Residents of Khan Yunis reported intense gunfire between Israeli troops and Hamas fighters. Elsewhere in Gaza on Saturday (December 30), fighting was also reported in the Nusirat refugee camp, Maghaji and al-Burej.
In a statement at X, the Israel Defense Forces (IDF) Arabic-language spokesman warned that Gaza's main north-south road, Salah al-Din, was a "war zone".
According to the Hamas-run Health Ministry, more than 21,672 people have been killed since the Israeli offensive began.
The ministry said another 56,165 Palestinians were injured.
Meanwhile, the World Health Organization (WHO) has warned of the spread of infectious diseases across displaced communities in Gaza.
The head of the WHO Tedros Adhanom Ghebreyesus said the number of cases increased across Gaza between mid-October and mid-December.
He said, about 1 lakh 36 thousand 400 people have diarrhea, 55 thousand 400 people have lice and scabies and 126 people have been infected with meningitis in the United Nations shelters.