The clashes, which have raised fears of a worsening conflict in Libya, began after the renegade commander of the self-styled Libyan National Army (LNA), Khalifa Haftar, ordered his forces to seize Tripoli from the UN-recognised rival Government of National Accord (GNA) on April 4.
Maria do Valle Ribeiro called for a lull in fighting on Monday but acknowledged aid partners need to be prepared for clashes to continue.
"We are likely to see a continuation of hostilities for some time to come," she said at a press briefing in Tripoli. "For that we need to gear up and accelerate and increase our capacity to be able to respond."
The UN and its humanitarian partners have reached some 21,000 people so far. In an appeal last week, the UN's humanitarian affairs office (OCHA) asked for $10.2m to cover the aid response up to May 18.
With fighting into its third week, around 32,000 people have been displaced, according to the UN's humanitarian agency.
"We're also very concerned by the fact that displacement is continuing at an increasing rate every day," Ribeiro told reporters.
Fighting continued overnight on Sunday around Libya's capital as troops aligned to the UN-recognised government targeted Haftar's forces.
Al Jazeera's Mahmoud Abdelwahed, reporting from Tripoli, said forces loyal to the UN-recognised government "started a new move this [Monday] morning towards the locations of Haftar's forces in and around the inactive Tripoli international airport on the southwestern part of the capital".
"In Tripoli, artillery fire was heard clearly overnight [Sunday] and government sources say that they were using heavy weapons to target Haftar's forces in the vicinity of the airport," he said.
Heavy clashes near Libya's Tripoli amid 'new phase of attack'.
"Humming of warplanes was also heard overnight, but no air strikes were reported," Abdelwahed added.
Haftar's LNA said on Monday it would intensify an assault on Tripoli.
Forces loyal to Tripoli drove back the LNA in recent days in the southern suburb of Ain Zara, the main scene of the fighting, Reuters reporters visiting the area said, even though the LNA said it had launched air strikes on military sites in the capital.
LNA spokesman Ahmed Mismari denied there had been a retreat but said an advance by his forces had slowed because of the dense population in the areas where fighting was taking place.
Fighting was under way on Monday for control of Ain Zara and Azizyia, two larger towns near Tripoli, and in the Abu Salim district, about seven kilometres (four miles) from Tripoli's centre.
The death toll from this month's fighting climbed to 254, including combatants and civilians, the World Health Organization (WHO) said on Sunday. At least 34 more people died in the past two days, WHO said, while 1,228 were wounded.
Fighting over Tripoli has spiked since the White House said President Donald Trump spoke to Haftar last Monday.
The disclosure of the call on Friday and a US statement that it "recognised Field Marshal Haftar's significant role in fighting terrorism and securing Libya's oil resources" has boosted the commander's supporters and enraged his opponents.
Since launching his push, which many see as a power grab for Tripoli, Haftar's forces have captured the districts of Gharyan and Qasr Bani Ghashir along with several smaller towns. They also seized the capital's shuttered old airport.