The Senate Armed Services Committee yesterday voted 11 to 6 to continue producing the MX missile as President Reagan mounted a major personal lobbying effort to win what White House officials called a "make-or-break" vote on the MX today in the Senate.

As the sharply divided Senate began debate on releasing $1.5 billion to produce 21 more missiles, the president telephoned at least one wavering senator from Air Force One and scheduled a trip to Capitol Hill to meet with Senate Republicans today on the MX and budget issues.

Although the lopsided vote from the strongly pro-defense Armed Services panel came as no surprise, it set the stage for the first major test of the administration's clout on Capitol Hill since Reagan's landslide reelection victory last fall.

Meanwhile, the Council of United Methodist Bishops joined Roman Catholic bishops who blanketed the Capitol with letters late Friday opposing the missile. The Methodists yesterday expressed strong opposition because they say the weapon would "increase the likelihood of nuclear war."

At the same time, a coalition of about 20 organizations opposed to construction and deployment of the MX met to coordinate their lobbying campaign in the House and the Senate. Some have set up telephone banks to call members to generate letters and telephone calls to the offices of senators or representatives regarded as undecided.

"We've got lots of talent in Washington," one anti-MX lobbyist said. "But our base is grass-roots activity. We cover all the bases here, but it's going to come down to the folks out there and the White House and Senate Majority Leader Robert J. Dole here."

Reagan has called the MX the "Peacekeeper," made it the centerpiece of his strategic arms buildup and termed it crucial to arms control negotiations now under way in Geneva. Opponents say the missile is excessively costly and vulnerable and would contribute to an unnecessary escalation of the arms race.

"This vote is absolutely, for us, a make-or-break vote," said presidential spokesman Larry Speakes. The White House is hoping a large victory today will provide the momentum to survive three other votes over the next week or so that are necessary to nail down MX funding for this year.

Leaders on both sides of the MX issue said they thought that today's Senate vote would be close.

Sen. Dole (R-Kan.) said he was "confident" that the missile would survive, but acknowledged that up to a dozen senators, half of them Republicans, were still undecided.

"I think we have it by one or two votes," said Armed Services Committee Chairman Barry Goldwater (R-Ariz.).

Some lobbying efforts were politely deflected. Senate Minority Leader Robert C. Byrd (D-W.Va.) turned aside lobbying efforts from Defense Secretary Caspar W. Weinberger and White House officials, who asked Byrd if Reagan could stop by to see him after his meeting with Senate Republicans today. Aides said Byrd told the officials that he would be glad to see the president after all the MX votes were taken.

Many who were planning to vote for the MX this time served notice that they would balk at the administration's request for 48 more of the missiles next year, at an estimated cost of $3.2 billion.

Among them was Sen. Sam Nunn (D-Ga.), influential ranking Democrat on Armed Services, who told his colleagues in opening debate that he would not "vote for anything approaching 48" more missiles. "I serve notice to the administration they have almost almost run my string . . . , " said Nunn, who has helped the administration preserve the MX up to this point.

With 21 missiles under production, the 21 additional missiles being voted on this week and 48 for next year would put the administration close to its goal of 100 deployed MX missiles.

Today's vote by the GOP-controlled Senate begins a week-long congressional obstacle course the administration must cover to release MX production funds that were frozen last year by Congress.

In all, four votes will be required under a complicated formula for release of the funds that Congress approved last year but froze until this month. The Senate is scheduled to vote twice this week, today to authorize the production and again on Thursday to appropriate the funds.

If the Senate approves the MX, the Democratic-led House will then vote on the same issues next week.

House leaders said yesterday the votes there will be close, with several dozen lawmakers not firmly committed. Democratic leaders began marshaling their forces yesterday to try to defeat the missile.

Votes on the MX were extremely close in both chambers last year. The House voted by margins of three or less to freeze the funds, and Vice President Bush had to break a tie in the Senate to save the missile.

Despite widespread predictions that the missile would be killed this year, its prospects improved dramatically when the United States and the Soviet Union agreed to resume arms control negotiations. The administration argued that defeat of the MX would undermine U.S. leverage in the talks, which began last week in Geneva.

Both supporters and foes of the missile agree that this argument has had considerable influence in Congress among lawmakers nervous about appearing to jeopardize the talks.

"I think the president may have an edge on this because of the bargaining chip he's talked about for Geneva," said Sen. John Glenn (D-Ohio), an MX foe. "There has been a shift in favor of the MX," said Sen. Paul Simon (D-Ill.).

As Reagan started lobbying senators while returning from a summit conference in Canada, Bush warned that a vote against the missile would "cripple" U.S. arms negotiators.

In a speech to the Baltimore Council on Foreign Affairs, Bush said, "To vote down the MX at this critical juncture would send the worst possible message to Geneva."

If Congress "pulls the rug out from under our negotiators . . . the damage to our hopes for successful arms talks may be irreversible," he said.

Bush also said rejection of the MX would "gravely weaken our national defenses; waste the billions already spent on the MX program; undercut our allies who have already stood firm in accepting new Pershing and cruise missiles; cripple the position of our negotiators in Geneva; and show the Soviets that, despite the progress our country has made, at a moment of historic importance some in the Congress of the United States still lack resolve."

Despite an intensification of administration lobbying for the MX, Goldwater said few votes appeared to be changing. Sen. Larry Pressler (R-S.D.), who had been counted by some as undecided, said he would remain an opponent. Earlier, Sen. David L. Boren (D-Okla.) announced that he was switching to support the missile.

His switch came despite efforts by some farm-state legislators, led by Sen. John Melcher (D-Mont.) and Rep. Thomas A. Daschle (D-S.D.), to link the MX to passage of farm credit relief legislation. Melcher and Daschle, who oppose the MX, said they are hoping to persuade other farm-state lawmakers either to oppose the MX or not to vote, in order to pressure the administration on the farm issue.

In opening Senate debate on the MX, Sen. Gary Hart (D-Colo.) attempted to counter arguments that killing the MX would weaken U.S. defenses, saying, "The MX weakens the United States because it makes us less secure" by inviting a preemptive strike by the Soviets.

Picking up the administration's argument about impact on the Geneva talks, Goldwater said congressional failure to continue MX production would be a "severe blow to our negotiating posture" at the arms talks.

Pre-vote tallies showed senators dividing evenly on the issue or slightly in favor of continued MX production. A survey by United Press International showed 44 senators supporting the MX, 41 against it, three leaning against and 12 who have not declared a position. The Associated Press showed the missile favored, 45 to 43. Assistant Minority Leader Alan Cranston (D-Calif.) tallied the Senate and found a 46-to-46 split.