Stock prices continued to climb today, after Monday's dramatic rally, but the Dow Jones Industrial Average closed below its high for the day.

"It was a continuation of a snap-back rally," said Donald McKee, a trader at the brokerage firm of Sanford C. Bernstein & Co.

However, McKee cautioned, although there is a "major psychological turnaround" in the market, the fundamental economic forces that have depressed stock prices since early summer remain.

Investors are concerned that President Reagan will not be able to control the federal deficit, which will lead to heavier-than-projected federal borrowing and continued high interest rates. Those high rates threaten to throw the economy into a serious recession, many investors feel.

White House spokesman David Gergen said, however, that the recent fluctuations in the stock market -- which fell 11 points Friday after the president's Thursday night economic address, then jumped Monday and today -- have nothing to do with investor attitudes toward Reagan's policies, but rather concerns about interest rates and corporate profits.

He also took a dig at Congress, saying there is investor skepticism about whether legislators have "the wisdom and courage to pass" Reagan's latest budget-cutting proposals.

The Dow Jones average, the most publicized indicator of stock price behavior, closed at 847.89 today, up 5.33 points from its Monday close. At one time in the early afternoon, the Dow average was up 11.89 points.

On Monday, after stock prices collapsed in Asian and European trading, the U.S. stock markets plummeted for about a half hour, then staged one of the strongest rallies in history. The Dow average was down more than 14 points at 10:30 Monday morning. It ended 18.55 points above its Friday close, nearly a 33-point swing, one of the largest in stock market history.

Meanwhile, the bond market declined today, after rallying along with the stock market Monday. Many bond investors are worried that because Congress has yet to pass the federal debt ceiling, forcing the Treasury to delay several planned securities sales, the government might have to sell a vast amount of debt in a short period of time to meet its cash needs, putting pressure on nongovernment bonds.

The worldwide collapse of stock prices was triggered by fears of higher interest rates and recession and by dire predictions by market analyst Joseph Granville that prices would plummet.

Today the rest of the world took its cue from Wall Street. In London, the Financial Times Index rose a record 23.7 points, and in Tokyo the Nikkei Dow Jones Index soared 320.56 points after falling 302.84 points Monday.

Trading was brisk on the New York Stock Exchange today as 50.3 million shares changed hands. On Monday, 61.32 million shares of stock were bought and sold.

Analysts said that while Monday's rally, and today's continuation of it, represent a climax to the price declines of recent weeks, it is premature to say that the stock market, where prices have been declining more or less steadily since the last week of April, has touched bottom. If history is any guide, there is likely to be a series of rallies and setbacks until the market establishes a firm "base" off which it will begin a sustained upward climb.

"The precise configurations of any market reversal is not important, and trying to mastermind it in advance is silly and dangerous and for scalpers only," said Phil Roth of the brokerage firm E.F. Hutton & Co.

Granville, whose predictions of a "Blue Monday" on Wall Street proved as exaggerated as reports of Mark Twain's death, today dismissed his inaccurate prediction and said on NBC's Today Show that "one day doesn't make any difference, but the trend will make all the difference in your life." He said he does not see a major "bottom" in stock prices around the world until "sometime in 1982."

White House spokesman Gergen said "we're pleased" that Granville's predictions did not come true.

Although the Dow average did not rise as dramatically today as it did Monday, more stocks on the New York exchange closed higher today than on Monday. About 1,200 stocks closed up today, compared with 911 on Monday, while only 365 declined today, compared with 659 the day before.

The NYSE's own index closed up 0.42 point at 66.85, while on the American Stock Exchange, the index was up 6.17 points to 291.96.

Traders said that the Dow closed below its highs of the day not because of any weakness, but because some investors chose to sell in order to make a profit.