The slow military approach may be helping Iraqi and coalition forces kill and capture more IS fighters but it has put trapped civilians at greater risk, according to residents who recently fled neighborhoods still in IS hands.
Lise Grande told the BBC residents were in grave danger as so-called Islamic State (IS) was directly targeting families.
Residents in the Old City sounded desperate in telephone interviews over the past few days. Islamic State also controls parts of Syria.
Earlier this month, coalition officials estimated the number of militants in Mosul to be under 1,000 - compared with 3,500 to 6,000 militants in and around the city before the offensive began last October.
In an area in western Mosul, a region taken over by Iraqi forces not long ago, there are signs of what it costs to get rid of ISIS: The destruction is everywhere, staggering and it stretches on for miles.
Iraqi planes dropped leaflets over the area Friday telling civilians to flee "immediately" to "safe passages" where they will be greeted by "guides, protectors and (transportation) to reach safe places", according to a government statement. The fall of the city would, in effect, mark the end of the Iraqi half of the "caliphate" declared almost three years ago by Islamic State leader Abu Bakr al-Baghdadi from Mosul. 'We're waiting for death at any moment, either by bombing or starving, ' he said, asking not to be identified for his safety. "Adults eat one meal a day, either flour or lentil soup".More news: OPEC deal to cut oil inventories, but shale key long-term
The secretary-general said that even though the United Nations is not present on the fighting settings to move terrorists away the west of the Iraqi city still has alarming reports on its crimes against population.
Iraqi Prime Minister Haider al-Abadi had initially hoped Mosul would be retaken by the end of 2016. Despite this, they continued their campaign of suicide vehicle bombings and snipers placed on rooftops to making the fighting hard in the neighborhoods they control, already a challenge because of the narrow streets and dense civilian population. People were seen collecting wild mallow plants in abandoned lots and also eating mulberry leaves and other types of plants.
It has also been a campaign marked by brutality and atrocities committed by Mosul's would-be saviours, including units from the Iraqi army, special police units and sectarian Shia jihadist groups fighting under the banner of the state-sanctioned but Iran-controlled Popular Mobilisation Forces (PMF).
Iran is a supporter President Bashar al-Assad's government in Syria.
Linking up the two sides would give Assad a significant advantage in fighting the six-year rebellion against his rule.