CANBERRA, Jan 31 (Reuters) - Chicago wheat futures fell on Wednesday and headed for their first monthly decline since September, as falling Russian export prices and a strengthening dollar weighed on U.S. prices.

Soybean and corn futures also dipped and were set for monthly falls after rain boosted prospects for crops in Brazil and Argentina at a time when markets are well supplied.

The most-active wheat contract on the Chicago Board of Trade (CBOT) was down 0.5% at $6.02-3/4 a bushel by 0604 GMT but down around 4% this month and not far from last September's three-year low of $5.40.

The market has been awash with cheap wheat from Russia, where export prices fell again last week.

A strengthening of the U.S. dollar in January has meanwhile made U.S. agricultural products less attractive to importers.

Demand for wheat is weak in China and elsewhere but with traders holding a big net short position, there's room for volatility over the coming months as the market waits for northern hemisphere harvests, said Rod Baker at Australian Crop forecasters.

Prices of wheat, soybeans and corn rallied on Tuesday but traders said this was bargain buying not backed by news about supply or demand.

Meanwhile, Argentine farmers have closed the 2023/24 wheat campaign with a harvest of 15.1 metric tons, slightly lower than initially hoped for but up nearly a quarter from last year, the Buenos Aires grains exchange said.

Brazil's January wheat exports are expected to grow 5.2% compared with a year earlier, the highest monthly volume in more than a year, data from grain exporters group Anec showed.

Condition ratings for winter wheat improved during January in Kansas, the top U.S. winter wheat producer, though ratings declined in other states including Texas, the U.S. Department of Agriculture said.

In other crops, CBOT soybeans fell 0.5% to $12.13-1/4 a bushel and were down around 6.5% in January and corn slipped 0.2% to $4.47 a bushel was around 5% lower over the month.

Soybeans hit a two-year low of $11.88 on Tuesday and corn has hit a three-year low of $4.37 twice this month.

Soybean oil and meal prices on China's Dalian Commodities Exchange have fallen as demand slows in China, the world's top soy importer, due partly to a shrinking pig herd there.

Analysts expect, however, that data on Thursday will show that U.S. processors crushed more soybeans in December than in any month on record.

(Reporting by Peter Hobson; Editing by Rashmi Aich)