* Iron ore price offers drop by $1/tonne, demand weakens
* Mills waiting for China's next step to buoy economy (Updates rebar price)
By Manolo Serapio Jr
SINGAPORE, June 5 (Reuters) - Spot iron ore prices fell on Tuesday as Chinese buying lost steam, with market players waiting for more action by the world's top iron ore importer to boost its economy and lift demand for steel.
Prices of iron ore and Chinese steel rose last week for the first time in seven weeks, spurred by hopes of higher steel demand as Beijing moves to speed approvals of infrastructure investments and could unleash measures to stimulate its economy as the U.S. recovery falters and the euro zone is trapped in a debt crisis.
But price offers for imported iron ore in China fell by a dollar per tonne on Tuesday, according to industry consultancy Umetal, and a benchmark spot index slipped after rising more than 2 percent last week, as buyers turned scarce.
"Demand did not continue to pick up from last week. Steel mills are worried that demand may not pick up as people had thought," said a Beijing-based iron ore trader.
"Unless there's clear action taken by the government, I don't think the situation will change."
The price of steel billet in the Tangshan area in China's top steel producing province of Hebei dropped 20 yuan ($3.14) a tonne over the weekend, he said, after rising for most of last week.
But Shanghai rebar futures gained 1.3 percent to close at 4,084 yuan a tonne, rebounding in step with other commodities ahead of talks to sort out Europe's debt crisis.
Those mills that have bought iron ore last week may be done with restocking, and won't lift inventories unless signs emerge steel demand would recover, said the Beijing trader.
Iron ore with 62 percent iron content IODBZ00-PLT fell 1.1 percent to $134 a tonne on Monday, based on the Platts reference index. It rose 2.3 percent last week.
"We're bidding for some cargoes but not aggressively. We're not that confident to buy at the moment," said another trader in Shanghai, who has around 300,000 tonnes of iron ore stuck at Chinese ports waiting for buyers.
"Sales are very slow. It's hard to push buyers to accept prices."
Miner BHP Billiton sold a cargo of 62.7-percent grade Newman iron ore fines at $136.2 a tonne at a tender on Monday, lower than a previous $137 deal for the same grade last week, traders said.
Vale, the world's biggest iron ore exporter, is selling a cargo of 65.82-percent grade iron ore lumps at a tender that closes on Tuesday.
The Brazilian miner sold four cargoes, totalling more than 252,000 tonnes, on Monday, one of which, a 64-grade material, was sold at $136.15 a tonne, slightly lower than a prior sale, said the Shanghai trader.
Shanghai rebar futures and iron ore indexes at 0702 GMT
Contract Last Change Pct Change SHFE REBAR OCT2 4084 +51.00 +1.26 PLATTS 62 PCT INDEX 134 -1.50 -1.11 METAL BULLETIN INDEX 134.72 -0.75 -0.55
Rebar in yuan/tonne Index in dollars/tonne, show close for the previous trading day
($1=6.3645 Chinese yuan) (Reporting by Manolo Serapio Jr.; Editing by Ed Lane)