Four Virginia financial institutions have agreed in two separate and unusual transactions to purchase and exchange assets currently held by financially troubled savings and loan associations.

In the first of the transactions announced yesterday, Bank of Virginia of Richmond said it plans to buy four branch offices of Northern Virginia Savings and Loan Association for $7.75 million. Bank of Virginia said it will convert the S&L's offices in McLean, Bailey's Crossroads, Sterling and Bristol into bank branches.

Depositors' accounts in those branches will be taken over by the bank.

The deal is believed to be the first inter-industry transaction in which a commercial bank acquires assets of a savings and loan association without a merger being involved.

In a similar, but unrelated, transaction, Family Savings and Loan Association of Springfield said it has reached an agreement in principal to sell a little more than half of its assets to Investors Savings and Loan Association of Richmond.

The purchase would involve four offices and an amount of liabilities held by Family similar to the assets being sold, and a cash payment of $600,000 to be made by Investors.

Indeed, Northern Virginia S&L President Martin L. Schnider said his firm faces a profit squeeze that has plagued the industry, but he added that the transaction offers it a chance to solve its problems without government assistance.

Under terms of the proposal--which could be completed by the end of this month, according to Bank of Virginia officials--the bank will assume $67 million in deposits now held at the four S&L branches. At the same time, Northern Virginia will pay the bank an amount in cash equal to the deposits, minus the $7.75 million purchase price.

A spokesman for the bank stressed, however, that it will not assume any mortgages in Northern Virginia's loan portfolio.

S&L customer accounts that would be transferred to the bank will retain the one-quarter-percentage-point interest rate differential paid by savings and loans.

"It marks the first time we know of where a commercial bank has purchased the branch offices of a thrift institution and where both entities continue to operate as separate entitites," said Schnider.

If approval is granted by regulatory agencies concerned, the deal will help the $2.8-billion-asset Bank of Virginia penetrate the Northern Virginia market while providing Northern Virginia S&L with needed cash, said Bank of Virginia Chairman Frederick Deane Jr.

A Federal Home Loan Bank Board report on the performance of the nation's savings and loans during the second half of 1981 shows that Northern Virginia S&L lost $3.8 million. Six months earlier it lost $2 million. It's net worth at the end of last year was $3.9 million, the bank board report shows.

Family Savings and Loan lost $941,000 last year, the same report shows.

Grashaw said the proposed transaction with Investors will help Family by giving it a capital infusion of $600,000, while enabling it to sell a large portion of its portfolio of below-market-rate mortgage loans.

After the sale, Family will operate a single office in Springfield with assets of $30,000.

The transaction is subject to a definitive agreement and approval by the Virginia State Corporation Commission and the Federal Deposit Insurance Corp.