Stocks registered strong gains today, apparently encouraged by President Reagan's conciliatory remarks yesterday on the federal budget deficit and reports of possible increases in spot oil prices.

Volume on the New York Stock Exchange rose sharply to 56.8 million shares, more than 25 percent higher than yesterday's trading totals and the Dow Jones Industrial Average climbed 10.47 to 833.24, its highest finish in seven weeks since it closed at 833.81 on Feb. 14.

There were also about three times as many advancing stocks as declining issues on the New York exchange, as its composite index climbed 0.97 to 65.49. Several brokers reported significant block trading, another sign of possible market recovery.

Standard Oil of Indiana was the most active stock, closing up 1 1/2 at 38, as chemical industry and other petroleum-related issues joined in the rally. Superior Oil rose 2 to 28 1/2 and Getty Oil jumped 1 1/8 to 45 1/4.

Some market analysts suggested that any indication the administration would move to cut the federal deficit is bound to unleash considerable pent-up demand for stocks by investment funds. "That was the first time he Reagan showed he would be a little flexible," said Marvin Katz, senior trader for Sanford C. Bernstein and Co. Inc. "That speech didn't hurt at all."

Reagan told a press conference last night that he is "not inflexible" on federal spending issues and suggested that he would consider cuts in proposed military expenditures as long as those reductions would not harm the nation's effort to "rebuild" its armed forces.

But Victor Melone, chief investment officer for the trust department of Manufacturers Hanover Trust Co., said he hoped Reagan would not move to cut federal defense spending before Congress demonstrates a willingness to cut entitlement programs. "The last thing I want him to do is give away his trump cards," Melone said.

Wall Street is convinced that many institutions are sitting on cash reserves waiting for indications that the administration will move to trim the federal deficit.

Richard McCabe, vice president and manager of the market analysis department at Merrill Lynch, Pierce, Fenner & Smith Inc., said a survey recently completed by his firm indicated the ratio of cash to assets at 132 leading institutions rose from 12.4 percent in mid-December to 15.2 percent as of mid-March.

The market started slowly this morning but the rally grew as the day progressed. The American Stock Exchange index closed up 3.25 at 263.37.

Big Board volume reached 57.10 million shares, up from 43.30 million yesterday.

Nationwide turnover in NYSE-listed issues, including trades in those stocks on regional exchanges and in the over-the-counter market, totaled 67.15 million shares.

Standard & Poor's index of 400 industrials rose 2.12 to 126.35, and S&P's 500-stock composite index was up 1.83 at 113.79. The NAS-DAQ composite index for the over-the-counter market closed at 177.31, up 1.66.