CHICAGO, June 7 (Reuters) - U.S. wheat futures fell on Wednesday, with the most-active Chicago Board of Trade soft red winter wheat contract snapping a five-session rally that had pushed prices to their highest in nearly three weeks, traders said.

But concerns about export demand for U.S. supplies pulled wheat prices lower, with high-protein offerings notching the biggest declines.

"Does this market have enough to justify going higher when Russia continues to dump wheat on the market?" StoneX chief commodities economist Arlan Suderman said. "Russia still sets the world wheat market price."

Corn futures fell, with contracts that track the recently seeded crop posting the biggest declines on forecasts for some rains in key growing areas of the Midwest this weekend.

"It looks like things are changing for the wetter," Marex Capital analyst Charlie Sernatinger said in a note to clients. "And the dryness that we have been seeing is the anomaly."

Soybeans ended mixed, with the most-active contract firming as traders staked out positions ahead of the U.S. Agriculture Department's closely watched World Agricultural Supply and Demand Estimates report on Friday. But the rainy outlook weighed on new-crop soybean contracts, which closed in negative territory.

The benchmark Chicago Board of Trade July soft red winter wheat contract dropped 11 cents to $6.16-3/4 a bushel. K.C. hard red winter wheat futures were down 4.0% and MGEX July spring wheat was off 2.8%.

Wheat futures had showed some strength during the overnight trading session on support from concerns over safe shipment of Black Sea supplies and lower output in Australia.

"There are some supply issues emerging, but U.S. wheat is struggling to find business," said one Singapore-based grain trader.

CBOT July corn futures were down 3-3/4 cents at $6.04-1/4 a bushel and CBOT July soybean futures settled up 7-1/2 cents at $13.60-3/4 a bushel. (Additional reporting by Naveen Thukral in Singapore and Sybille de La Hamaide in Paris; Editing by Rashmi Aich, Subhranshu Sahu, David Gregorio and Cynthia Osterman)