Prudential Insurance Co. announced agreement today to acquire Bache Group Inc., one of the nation's largest brokerage firms.

Prudential, the nations's largest insurance company, is a white knight for Bache management, which has been fighting a potential acquisition by a trio of Canadian financiers.

Prudential insures about 50 million Americans through its life, health and other products, the company said.

Prudential agreed to acquire the nation's eighth biggest brokerage firm for $32 a share, a deal that could result in Prudential laying out $385 million in cash. Bache would become a subsidiary of the insurance giant under the planned deal. Bache and Prudential said the insurance company expects to make a cash offer to Bache shareholders within a few days.

James R. Dillen, senior vice president of Purdential, said the insurance company was asked by Bache's financial advisers, First Boston Corp., about two weeks ago if Prudential had any interest in acquiring Bache. Dillen said that the types of financial products that brokerage firms offer fit into a long-range plan that Prudential just completed.

Dillen said, however, that Bache would be operated as an arms-length subsidiary for the foreseeable future. Bache management would continue to operate the subsidiary.

Meanwhile, Canadian brothers Samuel, William and Hyman Belzberg, who have acquired 22.6 percent of Bache's 11 million shares, had no comment on the Prudential-Bache announcement, a spokesman said. Samuel Belzberg, who has been directing the stock purchases from his Vancouver headquarters, has been in Europe since Wednesday.

Whether the brothers will make a counteroffer for Bache or merely collect about $40 million in profits by selling their shares to Prudential is anyone's guess, sources close to Bache said. About 40 percent of Bache stock is owned either by current management or by employes and others considered friendly to management.

Bache Chairman Harry A. Jacobs Jr. earlier denied a Belzberg request for two seats on the board, citing a Treasury Department report that Hyman Belzberg was seen in the company of underworld figures in 1970 at the Acapulco Hilton. At that meeting, a number of organized-crime figures such as Miami's Meyer Lansky and Montreal's Benny Kaufman met to discuss infiltration of gambling casino businesses in Atlantic City, N.J., according to government reports.

Bache asked the attorney general of New York to investigate the Belzberg purchases and issue an order prohibiting them from buying any more shares.

Soures close to the Belzbergs say that they were angered by Jacobs' tactics, but whether that will impel the Belzbergs, who have been buying Bache stock through their First City Financial Corp., to make counteroffer to the Prudential offer is not clear. "They are businessmen first," said one banker who knows their operation well.

Banking and financial sources in the United States and Canada say that the Belzbergs have an impeccable reputation for honesty and feel that Bache's moves, in the words of banker, wre "dirty pool."

In a statement, Jacobs and Prudential Chairman Robert A. Beck said the friendly merger "should prove an excellent short- and long-term venture for both companies." The Bache board of directors voted unanimously to approve the merger and urged all Bache shareholders to sell to Prudential when it makes the offer.

Dillen, of Prudential, said that while the Belzberg stock purchases were one factor in Bache's decision, Prudential also can be a source of new capital for Bache. Bache, once the second-biggest brokerage firm in the country (its brokerage subsidiary is Bache Halsey Stuart Shields Inc.) has slipped to eighth in the last year. Nevertheless, it runs one of the largest retail brokerage operations in the country, with more than 200 offices. Most of them are concentrated in large metropolitan areas.

Although Bache and Prudential negotiated in secrecy for two weeks, rumors of the impending merger hit Wall Street late Wednesday, and Bache stock climbed nearly $4 a share that day to $29.75. Bache did not trade today.

Bache directors were summoned to a meeting a 4 p.m. Wedensday, a spokeswoman for Bache said, but the directors were not told the purpose of the meeting. Bache and Prudential officials signed off on the friendly acquisition at 2:30 p.m. today at Prudential's Newark headquarters.

Whether the merger, if successful, will propel other insurance companies into the brokerage industry is not clear, according to Perrin Long of Lipper Analytical Services. But Long said the merger should be "a plus for Prudential. It expands their base of financial products for their policyholders and anybody else, for that matter."

Nearly all so-called financial services companies -- from brokers to banks to insurance firms -- have been exploring ways of broadening the services thay can offer clients and customers, in part to even out earnings and revenues. Bache, for example, already is the 10th-biggest insurance broker in the country. Banks have been trying to offer mutual funds.

Lathough insurance companies have been involved with brokerage firms in the past, nearly all those links have been broken in recent years. About a decade ago Prudential tried to buy a seat on the New York Stock Exchange in order to save commission fees on the large volume of securities transactions it performs regularly.The NYSE would not permit Prudential to join the exchange, and the elimination of fixed commission rates in 1975 presumably reduced the desires of big traders such as Prudential to have seats on an exchange.

Dillen said that Prudential has dealings with many brokers and that today's action would not change Prudential's approach to investments.