* Benchmark iron ore index at over 1-week top
* Chinese mills restocking ore after last week's price drop
* Traders wary of taking positions on shaky steel demand view
SINGAPORE, June 13 (Reuters) - Bids for spot iron ore cargoes in top market China edged up on Wednesday, with more steel producers keen on replenishing stockpiles after a recent drop in prices, although traders doubt the upward momentum can be sustained.
Chinese steel mills could curb output in response to slackening demand, with traders saying some smaller mills have already begun cutting production, and that may hurt demand for iron ore. China buys around 60 percent of global iron ore output.
"Buying inquiries have increased, and as a result prices have gone up slightly," said an iron ore trader in Shanghai.
"If you put your cargo in the market there'll be some people willing to bid for the cargo, which was missing completely last week."
Buyers are willing to pay $125 a tonne for 58-percent grade Australian Yandi iron ore fines, said a Hong Kong-based trader, up by a dollar from earlier this week.
There is also buying interest for higher-grade Australian 61.5-percent Pilbara fines at $132 per tonne, he said, up slightly from last week, but still well below current market offers of $134-$136.
"There's buying interest, but prices are still low," said the Hong Kong trader.
With steel prices in China down around 2 percent this year, mills are finding fewer reasons to continue producing at a record pace as they have since March, threatening the current upturn in iron ore prices.
The country, which produces around half of the world's steel, made 61.234 million tonnes of crude steel in May, up 2.5 percent from a year ago. But the annual rise was smaller than the 4 percent increase in March.
Benchmark iron ore with 62 percent iron content.IO62-CNI=SI rose 0.6 percent to $133.10 a tonne on Tuesday, according to the Steel Index, marking its third straight day of gains.
A fall in the price to a two-week trough of $130.60 last week drew buyers back into the market, although further gains are in doubt in the face of sluggish Chinese steel demand.
China's Baoshan Iron and Steel, the world's No. 3 steelmaker, has said it will cut prices of its main products by around 4 percent in July, its first reduction this year, reflecting its concern demand may not pick up soon.
"The fact that Baosteel's prices for July have been cut tells you the iron ore rally cannot last for too long," said the Shanghai trader, adding that steel mills were mostly driving the market higher because traders were wary about taking positions.
"Most traders are doing pure back-to-back and if there's a mill supporting them at a certain price they would definitely bite the bait, but not taking positions," he said, referring to the market term for traders buying on the spot market to meet clients' orders.
Shanghai rebar futures and iron ore indexes at 0705 GMT
Contract Last Change Pct Change
SHFE REBAR OCT2 4102 +2.00 +0.05
PLATTS 62 PCT INDEX 135.5 +1.25 +0.93
THE STEEL INDEX 62 PCT INDEX 133.1 +0.80 +0.60
METAL BULLETIN INDEX 134.49 +0.19 +0.14
Rebar in yuan/tonne
Index in dollars/tonne, show close for the previous trading day