Senate Majority Whip Alan Cranston (D-Calif.) cautioned President Reagan yesterday not to assume that his next nominee to the Supreme Court will receive less scrutiny than Robert H. Bork has faced in his apparently unsuccessful nomination.

"I don't think anyone should share the illusion . . . that the next nominee will sail though like a greased pig no matter what that nominee may be like," Cranston said. "We will take a hard look at the nominee."

Cranston's remarks came on ABC's "This Week With David Brinkley" shortly after Washington lawyer Lloyd Cutler, a Democrat and a Bork supporter, said he believed that the Senate would be less likely to vote against the next Reagan nominee even though the candidate might be more conservative.

Cranston said, however, that "there will be grave problems again" if the president selects "an extremist," someone with "erratic" opinions or someone who, like Bork, expresses doubts about a constitutional right to privacy.

At one point in the program, Cranston and Senate Minority Leader Robert J. Dole (R-Kan.) suggested that the Senate might take up the Bork nomination as early as Tuesday, a step that would require unanimous consent of the Senate.

"We're not going to win the Bork nomination," Dole said, urging expedited consideration of the nomination. "We're going to convince a lot of Americans, though, that the {nomination} process was not fair. Judge Bork was improperly treated."

Initially it appeared that the nomination of Bork, who sits on the U.S. Court of Appeals for the District of Columbia, would not come before the Senate until Oct. 20. That date was based on an agreement between Democrats and Republicans on the Judiciary Committee that the minority report did not have to be filed with the Senate until Friday.

A Senate Judiciary staff member said yesterday he understood the GOP report might be ready by late Tuesday, the same time the Democrats hope to complete their report.

Under Senate rules such reports are not subject to floor action for three days after they are filed, but the limit can be waived if the Senate agrees unanimously.

Both Cranston and Dole said they would voice no objections to moving more rapidly. A Senate Democratic aide said later that such a step would be consistent with the thinking of Majority Leader Robert C. Byrd (D-W.Va.) and other key Democrats.

Dole said he would agree to move this week with the assumption that a vote on the nomination would not come until late in the week, after several days of debate. Republicans are anxious, he said, to spell out how they believe the Democrats have conspired to kill the nomination for political purposes.

Dole said he did not agree with Reagan's assertion last week that Bork had come under attack by a "lynch mob." Dole said he would not go that far, but said he was troubled that some Democrats appeared to have decided to vote against Bork before all the evidence on his record was before the Judiciary Committee.

Dole said he too believed Reagan's next nominee will be confirmed and will not face the same campaign in the communications media that Bork faced.

"The next nominee's going to be a conservative, let's face it," Dole said. He will be "someone who believes in the same things . . . Judge Bork believes in and I believe that person will be confirmed."

Cranston rejected the idea that the media campaign killed Bork's nomination.

"It was Judge Bork's record and, finally, it was his performance before the Judiciary Committee. When he stated before the committee that he found no right to privacy in the Constitution, I think that one thing finally did him in."

Last Friday, although his nomination appeared doomed, Bork told the president that he does not want it withdrawn and expressed hope that the floor debate on it would help clear the public's perception of his record. Fifty-three senators have publicly expressed opposition to his elevation to the Supreme Court, enough to assure its defeat.