The Supreme Court nomination of Judge Robert H. Bork was all but buried yesterday by the announcements of 10 more senators that they will vote against confirmation, but conservative Republicans urged the embattled nominee not to give up.

With pressure building on him to ask President Reagan to withdraw the nomination, Bork and his wife, Mary Ellen, met with 16 GOP senators in the office of Minority Leader Robert J. Dole (R-Kan.). Dole said Bork did not indicate his intentions and said only that he wanted to discuss his apparently hopeless situation with his family.

Outside the Capitol, a scene resembling a pep rally for Bork unfolded as the senators escorted him to his car. When a reporter shouted to Bork, "Are you going to hang in there?" Bork did not reply but the senators began applauding and gesturing that he should fight on.

Meanwhile 10 more senators, including Florida Democrats Lawton Chiles and Bob Graham, who both had been considered undecided, and five other southern Democrats announced yesterday that they will oppose Bork's nomination.

Earlier yesterday, Bork met at the White House with Reagan, Vice President Bush, Attorney General Edwin Meese III and White House chief of staff Howard H. Baker Jr. White House spokesman Marlin Fitzwater said Reagan told Bork he will "stand behind him." And other officials said a national television speech by the president, which had been shelved earlier this week, is again being considered, possibly for the eve of a Senate vote.

"The president isn't backing down," one official said. "He's willing to go all the way. And that's what will happen unless Bork himself takes a different view."

Bork thus appeared caught in a political twilight zone, with Reagan and conservative GOP senators urging him forward in a cause that White House officials and other Senate Republicans privately say is lost.

After their show of support outside the Capitol, the Republican senators indicated by their tone that a showdown on the Senate floor over the Bork nomination could be long and bitter.

In an apparent reference to Senate Judiciary Committee Chairman Joseph R. Biden Jr. (D-Del.) and ranking committee Democrat Edward M. Kennedy (Mass.), Sen. Phil Gramm (R-Tex.) said, "The American people know what the people who cheated in college think. We want to hear from the straight-A student."

Biden has acknowledged he plagiarized from a law review article while in law school, one of the revelations last month that led to his withdrawal from the 1988 Democratic presidential campaign. Kennedy was expelled from Harvard University for cheating on an examination. He was later reinstated.

Gramm said the Bork nomination "deserves to be debated" by the full Senate.

"For the American people and the system, I hope we fight this out," he said. "I look forward to it."

Sen. Jake Garn (R-Utah) bitterly told a reporter: "I've never been this angry before. It's a blot on the Senate. I've voted for liberal judges before. That will never happen again. If they {Democrats} win {the White House}, there will never be another liberal judge as long as I'm in the Senate. The whole damn Senate will be tied up."

Despite the defiant tone of the conservatives, Republican strategists said privately that the Bork confirmation fight has been lost. "I'd say it's gone. And I don't see how they turn it," one GOP aide said.

Asked whether anyone in the meeting had told Bork he could still win confirmation, Dole said, "We told him we're still counting."

Beside Chiles and Graham, the other senators who yesterday announced their opposition to Bork were: J. James Exon (D-Neb.), Alan J. Dixon (D-Ill.), John B. Breaux (D-La.), Richard C. Shelby (D-Ala.), Wyche Fowler Jr. (D-Ga.), Jim Sasser (D-Tenn.), Albert Gore Jr. (D-Tenn.) and Kent Conrad (D-N.D.). Sen. Wendell H. Ford (D-Ky.) announced his opposition Tuesday night.

Sen. Mark O. Hatfield (R-Ore.), charging that Bork is the victim of a "lynch mob," said he would vote for confirmation.

Exon said Bork "has not exhibited due judicial temperment by his adamant position to fight to the end to certain defeat on the floor . . . .

"He appears incapable of grasping that there is a time to aspire and a time to concede," Exon continued. "I respectfully urge Judge Bork and the president to come down from their defiant mountaintop and end the confrontation."

In one of the toughest indictments of Bork, Graham said, "His scrupulous inattention to the human consequences of adjudication is seen as a threat by many who have only recently won -- hard-won -- progress toward equality for all Americans . . . . Scholastic brilliance, when it is bloodless and abstract, is nothing more than brilliance with blinders on."

Senate Majority Whip Alan Cranston (D-Calif.) said there are now 56 votes against Bork, 40 for and four senators undecided. Cranston also said that moderate Republicans are most eager to avoid being forced to vote on the divisive nomination.

"It's Republicans who don't want to walk the plank," he said. "They feel it's a loser either way. If they're loyal to the president, they get hurt. All the public opinion polls show that."

Senate Majority Leader Robert C. Byrd (D-W.Va.) said Bork would be "the final victim" of a GOP attempt to prolong the confirmation fight and force a floor vote. "I hope we will all consider Mr. Bork and his feelings and the fact that the court has a vacancy," Byrd said.

As he was leaving the Capitol, Bork was asked whether he planned to ask Reagan to withdraw the nomination. He flinched slightly but did not respond.

Dole said Bork "feels much better about it" after meeting with the 16 Republican senators. "If he was on the fence when he walked in, I think he's more determined to hang in there," he said.

But Dole added, "He knows it's his call."