BEIJING, June 16 (Reuters) - Chinese steel consumption remained high in the first five months of the year, with strong fixed asset investment helping to offset weaknesses in consumer demand, the country's steel association said on Thursday.

However, the old problem of overproduction, together with high raw material costs, continued to weigh on the sector, the China Iron and Steel Association (CISA) said in its latest monthly report.

"The market has entered its busy season, but affected by relatively high steel output, a slowdown in demand growth as well as weaknesses in the international market, steel product prices will remain in a volatile state in the near future," the report said.

The support provided by fixed asset investment was reflected by price movements in May, with CISA's index for long products used mainly in construction rising 1.56 percent over the month.

By contrast, CISA's flat product index -- covering steel products used in automobiles and household appliances -- crept up by just 0.21 percent over the period.

Leading steelmaker Baoshan Iron and Steel has already decided to cut its product prices by as much as 200 yuan per tonne in July as demand for its high-end steel products begins to fade.

Output of steel reached a record 60.25 million tonnes in May, according to figures released by the National Bureau of Statistics on Tuesday.

But in a sign that demand remained relatively healthy, product inventories continued to fall over the month, dipping 5.19 percent to 14.61 million tonnes, according to CISA figures.

Stockpiles had fallen to 14.43 million tonnes by June 10, 7.66 percent lower than the same time last year.

Average daily steel output stood at 1.922 million tonnes in the first five months of the year, up 8.5 percent compared to the same period of last year and equivalent to 702 million tonnes when calculated on an annualised basis.

CISA said rising energy prices and tighter monetary policies will put the steel sector under increasing pressure in coming months, and long-standing oversupply problems were likely to worsen as the peak steel consumption period passes.

It urged steel enterprises to "adjust production patterns and product structures in a timely manner" in order to ensure that prices remain stable in what is traditionally the off-season for the sector, beginning in July. (Reporting by David Stanway; Editing by Jacqueline Wong)