Dart Drug Stores Inc. is rushing to solve its financial troubles by July 1 to avoid defaulting on a $10.2 million interest payment that was due on its bonds four days ago, investment executives said yesterday.

"July 1 is the critical date" by which Dart hopes to reach an agreement with its bondholders and major creditor on the company's proposal to swap debt for stock, one bondholder said.

A source close to Dart said the firm is "on schedule" in efforts to complete the swap accord by July 1. Chances are "extremely remote" that it will have to default on the interest payment, the source added.

Faced with yearly interest payments of $28 million and a $32.4 million negative net worth, Dart is offering an unusual stock-for-debt deal as an alternative to filing for protection from creditors under Chapter 11 of U.S. bankruptcy law.

Under the company's latest proposal, each of its $1,000 bonds would be swapped for about $762 in new securities, consisting of a new, lower-interest bond and two classes of stock that together would amount to 75 shares of stock.

This plan, outlined to bondholders in a Manhattan meeting late Wednesday, represents a "20 percent" improvement from the initial exchange offer that Dart made in late April, according to a bondholder. Under that plan, a $1,000 bond would have been swapped for an even-lower-interest bond and 35 shares of stock.

"There was a unanimity of feeling at the meeting {of bondholders} that a deal is doable between now and July 1," said one bondholder who attended the meeting. "At this point, there is no reason to believe any of the parties want to see the company go into Chapter 11," he added. "There may be minor changes . . . some tinkering" before 85 percent of the bondholders agree, the bondholder said.

Dart has said that for the swap to succeed it needs acceptance by 85 percent of the bondholders and an agreement from its major creditor, Equitable Life Insurance Co. of the United States.

"If the exchange offer is not successfully consummated, the company may be forced to seek a reorganization under Chapter 11," Dart has said in filings with the Securities and Exchange Commission. "Under certain circumstances, the company may be forced into liquidation under Chapter 7 of the bankruptcy code," Dart added.

Many of Dart's problems stem from the $10.2 million interest payment it has to make twice yearly on the $160 million high-risk junk bonds issued last summer. The bonds were used to help finance the debt incurred when the company was bought by the drugstore management from Washington's Haft family for $160 million in 1984.

The possibility of going into technical default on July 1 "is not something that appears to bother bondholders, but Dart's trade creditors {suppliers} appear to have attached some significance to the July 1 date," the bondholder said.

Under the latest proposal, every $1,000 bond that now bears a 12.7 percent interest rate would be exchanged for a new 8 percent, $500 bond plus two forms of stock equal to 75 shares: 25 shares of common stock and five shares of preferred stock. Each share of preferred stock would be convertible into 10 shares of common stock. The common stock is valued at $3.50 a share.

The initial proposal would have exchanged each $1,000 bond for a 6.5 percent, $500 bond and 35 shares of common stock, worth $6.70 a share. The per-share value of the stock would have been higher under the initial proposal because there were fewer shares being offered.