United Virginia Bank said yesterday that it will ask E. F. Hutton & Co. for at least $350,000 to compensate for its losses in the brokerage firm's illegal check-kiting scheme.

Hutton pleaded guilty May 2 to 2,000 counts of mail and wire fraud, was fined $2 million, and agreed to create an $8 million reserve to make restitution to banks from which it improperly obtained interest-free loans in an 18-month period starting in July 1980.

The banks' deadline for filing applications was last midnight, but Hutton has granted extensions to several, including UVB. Some are said to be having difficulty reconstructing the transactions at issue.

"Hutton is not announcing the number of banks that have applied for restitution or commenting on their names or the amount of restitution sought," Hutton said. "The restitution program will continue for some time while Hutton and the banks analyze the claims." Spokesman Laurel Fry said the firm will announce an exact figure when it is available.

In June, the UVB case was one of two cited by the Justice Department "to show dramatically the essence and scope of the Hutton scheme." On Nov. 2, the department said, Hutton's branch in Alexandria deposited customer funds of $70,629 -- and took out 140 times as much. "The amount drawn from the account that day was $9,953,400," the department said. "The overdraft was covered with a zero balance account check."

Between January and December 1981, the same branch deposited a total of $3,537,397 in customer funds and withdrew from the same account $640,796,712, or "well over one-half billion dollars in overdrafts," the department said.

UVB will file its claim by early September. Only then, a spokesman said, will the bank know precisely how much it will seek from Hutton.