Directors of Virginia National Bankshares Inc., unanimously rejected today an offer by an unidentified European banking and investment group to invest about $20 million in new equity capital in VNB. The investment would have amounted to a 13 percent interest in the bank holding company.

"The board had some hesitancy in having anybody own 13 percent of the corporation," chairman W. Wright Harrison said of the two-week-old offer. "We don't really have any need for the money, so we decided to stay as we are."

The offer marker the first time that any European group came up with a concrete proposal for acquiring an interest in Virginia National. Harrison said, however, that internally generated capital would be adequate to support the corporation's future growth. He declined to reveal the identity of the group, saying it could harm negotiations with other banks in which the Europeans are interested.

[Separately, informed sources in Washington said the banking and investment group is based in Spain. Reportedly the group, has a number of banks on a list of potential investments.]

The announcement by Harrison came after VNB's annual meeting here, at which the bank holding company reported a sharp gain in profits.

Virginia National said that income before securities transactions was $5.4 million (76 cents a share), up 36 percent over $3.8 million (56 cents) in the first quarter of 1978.

Harrison noted that the robust quarterly earnings came on top of record earnings in 1978. As previously reported, Virginia National's earnings last year were $18.8 million before securities transactions, up 31.7 percent over 1977. Nationwide, bank earnings last year rose about 20 percent.

Hattison attributed the earnings surge to a "positive economic environment," which he said produced strong loan demand and rising interest rates.

But he cautioned that Virginia National won't be able to sustain the growth in first-quarter earnings for the rest of the year. Because of "uncertain economic conditions," he said, the banking firm has become increasingly selective in granting commercial credit.

In addition, Harrison said Virginia National Bank increased its interest payout on passbook accounts this month to 5 percent of the average daily balance, the maximum allowable by law. The bank, VNB's major subidiary, previously paid 4 1/2 percent and computed interest on the low monthly balance.

Harrison said he was initially "shocked" by reports that the Carter administration may ask the Federal Reserve Bond to boost interest rates dramatically to control inflation.

He said such Fed action could hike the prime rate as high as 14 percent, compared to 11 3/4 now, and bring about a recession as severe as the 1974-1975 economic slump.

"But in thinking about it, this move may be wise," Harrison said. "It would be terribly painful, but I don't see any other way to stop inflation."