The Food and Agriculture Organisation, World Health Organisation and the UN Children's Fund in Bangladesh said they were willing to offer the use of "internationally certified" laboratories to test for the industrial chemical.
Growing numbers of countries have pulled mostly China-made milk products from shelves after tests found they were tainted with melamine, which is normally used to make plastics and fertilisers.
Contaminated powdered milk has caused the deaths of four children and sickened more than 53,000 others in China in a weeks-long scandal.
Melamine is believed to have been added to milk in China to give it the appearance of higher protein content.
The three agencies said Bangladesh should remove all milk products from the market as soon as tests confirm their levels of melamine which can cause kidney stones and related illnesses in infants.
The statement comes two days after a top government official said results showing melamine in seven brands of powdered milk were "inconclusive and confusing."
Three major international dairy firms, Nestle, Arla - which produces Dano products - and New Zealand Dairy Products have challenged the Bangladeshi results and say their products are safe.
The initial tests, carried out by Dhaka University's chemistry department, were contradicted by results from two other local labs, which found melamine in only one of the brands tested.