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.

FUNDAMENTALS

* The most-active what contract on the Chicago Board of Trade (CBOT) was down 0.4% at $6.03 a bushel by 0104 GMT but down nearly 4% this month.

* CBOT soybeans fell 0.2% to $12.16-1/2 a bushel and were down more than 6% in January, while corn slipped 0.1% to $4.47-1/4 a bushel were around 5% lower over the month.

* Wheat is still a little way from last September's three-year low of $5.40, but 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.

* Prices of all three contracts rallied on Tuesday but traders said this was bargain buying not backed by news about supply or demand.

* Commodity funds hold sizable net short positions in all three markets, leaving them primed for short-covering rallies.

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

* The wheat market has been awash with cheap grain from Russia and Russian export prices fell again last week.

* Argentine farmers have meanwhile 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 soybeans, AgResource Company lowered its forecast of Brazil's 2023-24 soybean crop to 145.40 million metric tons, from 150.7 million previously.

* However, Brazil's harvest would have been smaller without recent rains, and Argentina is expecting a bumper crop to add to ample U.S. supplies.

* 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. soybean processors crushed more soybeans in December than in any month on record.

MARKETS NEWS

* Short-dated Treasury yields rose and a gauge of global equities seesawed near two-year highs on Tuesday after strong U.S. labour market data again underscored a resilient economy and raised questions about how soon the Federal Reserve will cut interest rates.

(Reporting by Peter Hobson; Editing by Rashmi Aich)