* China mills cut iron ore stocks to below 20 days - Macquarie
* Shanghai rebar has worst month in nearly a year (Updates price, add comments)
By Silvia Antonioli and Manolo Serapio Jr
LONDON/HONG KONG, Aug 31 Iron ore rebounded slightly on Friday but remained near a three-year low and posted its worst month since October 2011 as demand for new shipments from Chinese steelmakers remained weak.
The steel ingredient has been the hardest hit among industrial commodities by China's slowdown, losing 36 percent this year as demand fell after steel prices plunged.
Iron ore with 62 percent iron content, the industry benchmark, rose 0.8 percent to $89.40 per tonne on Friday, according to data provider Steel Index.
It was its first rise in two weeks, but left iron ore close to its lowest level in almost three years. In August prices fell more than 24 percent in their biggest fall since last October.
"The sentiment is still negative but it seems that the market is a little more steady than last week," said a UK-based iron ore trader, adding that iron ore offers from a major producer on Friday where at a similar price level to the two previous days.
"I think from now on the fall will slow up. It's a bit too early for a rebound but I don't think there will be drops as big and as sharp as last week, now."
Analysts said the price developments will also depend on the iron ore industry response to weak demand and lower prices.
"If everyone keeps producing iron ore we can expect to see a sharp fall in the short term before a rebound," said Hatch iron ore consultant David Tucker.
"My analysis suggests that we are already seeing a supply side response from Chinese domestic producers and this will increase if prices fall further."
Another key element is the steel production rate in China.
Crude steel output in the top producer and consumer of the alloy has hit record high levels this year but a steep decline in steel prices and profitability is now pushing mills to cut their production levels.
Data shows that China's average daily crude steel output fell 2 percent between August 11-20 compared with the first 10 days of the month and steelmakers have also cut their iron ore inventory levels.
Smaller Chinese steel mills had cut their inventories to about 17-18 days worth of consumption, versus 27-30 days over the past year, according to Graeme Train, analyst at Macquarie in Shanghai.
He said the "absolute minimum" inventory would be 14 days, which is about the period it takes for cargo imported from top exporter Australia to reach China.
"There's potential for the mills to destock further, so we could see prices go down to $75-$80," he said.
Shanghai steel rebar futures have dropped about 17 percent this year, less than half the percentage loss for iron ore.
Shanghai rebar fell nearly 10 percent this month, the steepest drop in almost a year, as losses extended to a fifth straight month.
In the face of weak steel demand, most Chinese mills have kept their iron ore inventories low, with some opting to buy small lots from port stockpiles instead of ordering fresh, big cargoes.
China's imported iron ore inventories fell slightly this week but remained at a high level.
Brazil's Vale SA, the world's top iron ore miner, said it was surprised that prices had fallen below $110 per tonne, which it attributed to excess supply rather than weak demand.
Jose Carlos Martins, who runs Vale's iron ore business, however, said the company's production costs were still well below current prices.
"We are one of the most efficient producers," he said. "We will be the last to leave the market."
(Editing by Jason Neely)