CHICAGO, April 22 (Reuters) - U.S. corn, wheat and soybean futures hit multi-year highs on Thursday as concerns about tightening global grain supplies triggered short-covering and fund-driven buying, analysts said.

Nearby Chicago Board of Trade corn futures closed up their daily 25-cent limit. Front-month CBOT wheat rose above $7 a bushel and soybeans above $15, both for the first time since 2014. CBOT corn, which surpassed $6 this month, hit its highest level since 2013.

CBOT benchmark July corn settled up 25 cents at $6.31-1/2 a bushel. July wheat ended up 35-1/2 cents at $7.10-1/2 after rising its 40-cent maximum, while July soybeans settled up 34-3/4 cents at $15.14-1/4 a bushel.

"These markets are largely over-bought and past due for a correction, but nobody wants to step in front of this train until the momentum turns," StoneX chief commodities economist Arlan Suderman wrote in a client note.

Firm domestic cash markets suggest that supplies left from the 2021 U.S. corn and soy harvests are dwindling, and traders remain concerned about a U.S. cold spell this week and dryness in Brazil threatening prospects for the next crop.

Wheat futures climbed on unfavourable weather in North America and the prospect of high corn prices boosting demand for wheat in livestock feed.

"This (rally) is because corn is in short supply and is likely to remain so in 2021/22," Commerzbank said in a note.

Chinese buyers are thought to have snapped up at least half a million tonnes of new-crop French wheat for shipment over July-September, which may be partly used in feed, traders said.

Support from low U.S. soybean inventories was amplified by tensions in related oilseed and vegetable oil markets.

Front-month CBOT soybean oil extended a rally to a near 13-year high of 62.69 cents per lb before settling at 62.52 cents, while palm oil reached a one-month peak. (Additional reporting by Gus Trompiz in Paris and Naveen Thukral in Singapore; Editing by Alexander Smith and David Gregorio)