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Ukrainian soldiers are still defending the industrial city, with heavy casualties on both sides.
Mr Putin’s military saw thousands of troops killed during his failed lightning invasion plan which started on February 24.
"On Russian fatalities, our figure is between 15,000 and 20,000 dead," said one western official.
"That is a change from our previous figure that we have been talking about in excess of 15,000 before."
But he stressed it was difficult to be more precise.
He added: "In terms of the munitions that we are talking about (running low), long-range precision missiles, advanced weapons, they take significant effort from a defence industrial complex to produce, and they are expensive."
While Russia had stockpiles of these weapons, they were keeping some in strategic reserve in case of a conflict with Nato.
"They are not necessarily running low on everything and of course some of the older munitions, and dumb bombs, they have very large reserves of and are using those in significant numbers as well."
According to the Western official Russia has been forced to deploy “several weapons systems which have been out of front line service for many decades”.
This, the official explained, included most notably, several battalions of 50 year old T62 tanks.
“This highlights how in some sectors Russian forces are deploying equipment which is more suitable for defensive rather than offensive operations,” the official said, adding that Russian forces were also having to pull ageing explosives and munitions out of deep storage.
“Because of the huge expenditure of munitions through this conflict they are absolutely bringing things out of deep storage and when you bring things out of deep storage you have to be very careful about what conditions it has been kept in but then also how you activate those munitions and make them fit for purpose. That process hasn’t always run smoothly for the Russians and explosives have a shelf life.”
As the war has gone on into its fourth month, both armies are under huge pressure.
"The Ukrainians and Russians face the same challenge," said the western official.
They are 100 plus days into a bitter conflict with significant fatalities and casualties on both sides.
"They will of course be facing morale issues.
"But the Ukrainians are fighting for their homeland and they are in largely well dug-in defensive positions so they have the advantage over Russians in that regard."
The Western official said he believed the key strategic city of Severodonetsk, which Ukrainian President Volodymyr Zelensky has said is vital to the battle for the Donbas, will fall to the Russians.
But he added Putin’s forces would still face challenges in making rapid advances in the eastern Ukrainian region.
“It’s a grinding military operation” and there will be a significant cost to Russian forces, he said.
While Ukrainian forces were also suffering heavy casualties, the Western official added: “It’s now also becoming clear but Russian campaign continues to be deeply troubled at all levels. Tactically morale at unit level and in some layers of command is probably dire.
“They are still struggling to work as a joint organisation taking days or weeks to achieve even modest tactical goals such as taking individual villagers, which doctrine suggests they should achieve.”