President Reagan yesterday opened a new drive to win congressional approval for 21 additional MX missiles, triggering a process that will lead to a vote on Capitol Hill about a week after the United States and the Soviet Union resume arms control negotiations in Geneva.

Reagan told a group of Senate Armed Services Committee and Appropriations Committee members that he is "utterly convinced we cannot get a sound agreement in Geneva" without congressional approval of the troubled missile program, a White House official said.

In the opening salvo of the administration's offensive, Secretary of State George P. Shultz and Secretary of Defense Caspar W. Weinberger said approval of the MX is linked directly to U.S. efforts to win deep arms reductions from the Soviets.

But congressional leaders said prospects for the MX remain uncertain. Senate Budget Committee Chairman Pete V. Domenici (R-N.M.), a supporter, said it "seems to me it's a tough case" in the Senate.

House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) said the vote will be "close" in each chamber. Private White House estimates are that Reagan faces a "very tough battle in the Senate," where last year Vice President Bush had to break a 48-to-48 tie, officials said.

Sen. Sam Nunn (Ga.), ranking Democrat on the Armed Services Committee, said recently that "sentiment is more favorable by a substantial amount than it was last fall" for the missile.

"Congress is always reluctant to kill a weapons system during arms control negotiations," he said. But pressure for defense spending cuts is working in the opposite direction, Nunn added.

Armed Services Committee Chairman Barry Goldwater (R-Ariz.), who once suggested that Reagan abandon the MX, more recently expressed reservations about doing so in light of the Geneva talks set for later next month.

Another Armed Service Committee ranking Democrat, Sen. John C. Stennis (Miss.), said yesterday that he has "serious doubts" about voting again to finance production of the MX.

At stake is $1.5 billion that Congress decided last year could not be spent for an additional 21 missiles without affirmative votes by both chambers this year. The missiles would be in addition to the 21 approved for production.

Reagan also is seeking funds for 48 more missiles in his fiscal 1986 budget to be debated later this year. Altogether, the administration plans to build 223 missiles and deploy 100 in existing hardened missile silos.

Under the process agreed to last year, money for the next 21 missiles was set aside until after March 1, and Reagan could ask for it by submitting a report to Congress on the MX and its relationship to arms control and national security.

The White House said yesterday that Reagan will submit the report Monday, the earliest opportunity to send it to Capitol Hill. This turns the issue over to the House and Senate Appropriations and Armed Services committees, which must bring it to the floor in no more than 15 days.

The key votes are expected to occur about March 20, eight days after arms control negotiations resume in Geneva. Reagan must win two separate votes in each chamber on the authorization and appropriations for the MX.

White House spokesman Larry Speakes said Reagan's goal was to get a vote before the Easter congressional recess. House Minority Leader Robert H. Michel (R-Ill.) said he discussed with Reagan yesterday "the orchestration" of votes in the House and Senate, with the Senate going first.

"We consider that to be of absolute paramount importance, to get the positive kind of vote there, particularly a week after those talks begin in Geneva," Michel said.

Michel said O'Neill had assured him that the House also would vote before the Easter recess.

Yesterday, the administration went all-out to use the Geneva talks as leverage to win political support for the 10-warhead intercontinental ballistic missile that has been the center of controversy for a decade.

The president, in the meeting with senators, said he expected that the Soviets will first offer at the Geneva talks to reduce the number of their older missiles, not their newer ones, a White House official said. Reagan added that the United States would then be in a position to reduce the number of its older missiles, and should have the newer MX to counter Soviet strength, the official said.

Asked whether the MX would be used as a bargaining chip in the Geneva talks, Speakes said that term has "such an erroneous connotation. We think that MX is an important part of our defense buildup."

But another White House official said the MX is "a powerful negotiating lever" to offset 600 Soviet SS18 and SS19 missiles in the same class as the MX.

Shultz said U.S. arms control negotiators should be sent to Geneva "with the strongest possible negotiating position . . . and that means not suggesting unilateral concessions that might diminish the incentives the Soviets have to talk."

Weinberger said the missile is "vitally important to our ability to achieve the deep reductions we seek" in Geneva.

"Why should they seek to reduce their arsenals," he asked of the Soviets, "if we have signaled we are going to permit them to maintain -- and perhaps even expand -- advantages they currently enjoy?"

White House officials said the report Reagan will send to Congress Monday will discuss advances in hardening missile silos and possible basing modes for the "Midgetman" single-warhead mobile missile planned for production after the MX.