* Iron ore market seen oversupplied, soft mill demand
* Prices rallied for 10 straight days to Thursday
* Some traders seen under pressure to keep prices up
SINGAPORE, June 25 (Reuters) - Spot iron ore prices may struggle to stretch gains this week, as supply outpaces demand with China's steel market staying sluggish, raising the risk for traders hoping to resume a recent 10-day rally.
Price offers for imported iron ore cargoes in China, the world's biggest buyer of steel's raw material, were steady on Monday, although major miners Vale and BHP Billiton are slated to sell cargoes via tenders, traders said.
"It seems we are in an oversupplied market but there are some people out there who have a lot of cargoes and are trying to keep the market artificially up," said a Singapore-based iron ore trader.
"If you're a trader and you have 3 million tonnes which you bought at $135 (per tonne) you have an interest to try and keep the market up to be able to sell those 3 million tonnes at a higher level."
Benchmark iron ore with 62 percent iron content .IO62-CNI=SI was unchanged on Friday at $137.40 a tonne, with China away on a public holiday, after a 10-day rally that pushed up the price to a six-week high on Thursday, based on data from Steel Index.
The 10-day rise was iron ore's longest winning streak since mid-November, when it climbed for 14 days in a row. Behind the rally were traders betting that high steel output in China will push mills back into the spot market to restock.
Iron ore prices usually rise along with steel prices in China. But the recent rally transpired even if Chinese steel prices remained largely weak amid soft demand in the world's top steel market, suggesting prices of the raw material may struggle to build on recent gains.
"If it's up to the mills, I don't think they're going to be very excited at the moment," said another trader in Singapore.
"It's still hand-to-mouth for a lot of them. So sooner or later, fundamentals will prevail and the market will inevitably come down."
The most-traded steel rebar contract for October delivery on the Shanghai Futures Exchange closed down 0.3 percent at 4,107 yuan ($650) a tonne. It hit a session trough of 4,095 yuan, its lowest since June 14.
Physical iron ore prices were largely flat, said a Hong Kong trader, ahead of sale tenders by Vale and BHP Billiton.
Brazil's Vale is offering 250,000 tonnes of 60-percent grade iron ore, while BHP Billiton will sell a combined 170,000 tonnes of 62.7-percent grade Australian Newman iron ore fines and
57.7-percent grade Yandi fines, traders said.
Shanghai rebar futures and iron ore indexes at 0702 GMT
Contract Last Change Pct Change
SHFE REBAR OCT2 4107 -11.00 -0.27
PLATTS 62 PCT INDEX 139.25 +0.00 +0.00
THE STEEL INDEX 62 PCT INDEX 137.4 +0.00 +0.00
METAL BULLETIN INDEX 138.18 +0.00 +0.00
Rebar in yuan/tonne
Index in dollars/tonne, show close for the previous trading day
($1 = 6.3642 Chinese yuan)