National Permanent Savings Bank is negotiating a takeover of troubled Family Savings and Loan of Springfield, Va.

The chiefs of both institutions confirmed the deal but declined to discuss details now being worked out with the Federal Savings and Loan Insurance Corp. Given Family's financial plight, the acquisition is likely to involve government assistance. Several other attempts in the past year or so to find a merger partner for Family have been unsuccessful.

Year-end figures from the Federal Home Loan Bank Board show Family suffered a loss of $1.6 million in the last half of 1982 and was left with a negative net worth of $1.4 million. Family's president, James M. Grashaw, said the S & L suffered smaller losses during the first half of this year, but he declined to provide data.

Like many S & Ls, Family got caught in an interest-rate squeeze when the yield on its loans failed to match what it was paying out on savings. However, according to sources, Family failed to recover from the recession because it did not make a sufficient number of high-interest loans during that period to raise the average yield on its loan portfolio when rates declined.

Founded in 1973, Family also has consumer branches in Annandale, Fairfax, Reston and Burke. Three others were closed last year. Grashaw said National Permanent hopes to find jobs for all Family's 59 employes.

Since Family is a federal stock asociation, the acquisition will be put to the shareholders. An analyst with Lang & Co., which made a market in Family stock until a few months ago, said the value of Family stock was around $1 per share. He said National Permanent is offering shareholders a $3 cash payout, based on performance, over many years.

National Permanent, which transformed itself from a savings and loan into a mutual savings bank earlier this year in order to expand, is also planning a merger with Suburban Savings and Loan of Annandale. The bank's president, Edgar Peterson, called the two transactions entirely separate and denied that FSLIC had pressured National Permanent to take over Family as a condition for approving the Suburban acquisition.

In related news, First Virginia Banks announced recently it plans to acquire Peoples Trust Bank of Exmore, Va. Shareholders will receive six shares of First Virginia for each share of Peoples Trust. It will become part of First Virginia Bank-Eastern Shore. The agreement is subject to stockholder and regulatory approval.

First Virginia also plans to purchase the Williamsburg office of Virginia National Banks, which is required to divest itself of the office as a condition for regulatory approval of its merger with First & Merchants Corp.