Some farmers are being paid less than the cost of production, the National Farmers' Union says.
Protests have included removing large quantities of milk cartons from shops and blockading distribution centres.
Supermarket chain Morrisons will meet farming industry leaders on Tuesday to discuss the price row.
Both meetings will include the UK's four main farming unions: the NFU, NFU Cymru, NFU Scotland and the Ulster Farmers' Union.
"The situation many of our members are experiencing has become a crisis," NFU president Meurig Raymond said.
"In dairy, many milk producers have seen price cut after price cut.
"It's simply not sustainable for any farmer to continue to produce milk if they're selling it at a loss."
On Sunday two cows were taken down the aisles of a supermarket in Staffordshire as part of a protest by about 70 farmers to raise awareness about the situation.
Asked by the BBC if this protest was "responsible", Mr Raymond said: "I urge my fellow farmers, do not alienate our British customers at this given time.
"Keeping the British consumers on side is so important."
Arla, Britain's biggest milk co-operative, previously announced a price cut of 0.8p per litre - taking the standard litre price to 23.01p for its UK members.
Meanwhile, British dairy organisation AHDB Dairy said the the average UK farm gate price was 24.06p per litre in May, a decrease of a quarter over 12 months.
Farmers estimate that it costs between 30 and 32 pence to produce each litre of milk.
Protesters taking part in the so-called "Milk Trolley Challenge" have been removing all cartons of milk from shops including Morrisons and Lidl before paying for it and taking it away or dumping it at the checkout.
Ahead of the talks on Tuesday, a spokesman for Morrisons said: "We want to reiterate that we are not seeking any further reductions in milk prices and we will continue our talks with the NFU, in a constructive manner, to finalise our agreed plan of action."
A spokeswoman for Arla said the co-operative was "acutely aware" of the difficulties farmers are facing.
"We are doing everything possible to help our farmer owners to navigate through this increasingly tough situation, in the best possible way," she said.
