Supporters of embattled CIA Director William j. Casey yesterday launched a public campaign to help him keep his job, while the president offered to stron;ger defense of his intelligence director and old friend.

Three prominent friends of Casey, including Sen. Paul Laxalt R-Nev.), called a press conference to defend the man they say has distinguished himself in business, politics and government and to counter demands that he resign in the wake of his clandestine service chief, Max Hugel, being forced from office by allegations of financial wrongdoing.

Almost atthe same time, White House spokesman David R. Gergen said that "the president believes that Bill definitely ought to remain on the job."

Gergen called the Saturday briefing to deliver the new hard line on Casey, after the unexpected statement Thursday night by Sen. Barry Goldwater (R-Ariz.) that Casey ought to resign because of the poor judgment he showed in appointing Hugel to one of the most sensitive jobs in government.

The president, Gergen said, "believes that Mr. Casey is doing a fine job at the CIA. In light of that background the president is standing fully behind Mr. Casey." Gergen added that Reagan has received calls from "three to five" senators urging him to "withhold judgment until all the facts are in and have been carefully weighed."

The message was similar a few blocks away at the Mayflower Hotel where Leonard Marks, former director of the U.S. Information Agency; Stanley Sporkin, CIA general counsel and former enforcement chief at the Securities and Exchange Commission, and Laxalt praised Casey's character and service.

Sporkin, who said he is involved in Casey's defense because the Senate Intelligence Committee has directed questions to the CIA general counsel's office, also denied allegations that Casey misled potential investors in a corporate fund-raising effort in 1971 and that fugitive financier Robert Vesco had been involved in Casey's past business enterprises.

"Believe me," said Sprokin, "I know securities fraud when I see it and this isn't it." It is a "horrendous" injustice "talking about conduct that took place in 1968 and 1969," Sporkin said, referring to recent court rulings that have criticized Casey and other directors of a large-scale farming venture, Multiponics Inc.

When a reporter quoted from a New York federal judge's ruling that :Casey had "misled" investors in the Multiponics case, Sporkin fired back, "Now why do you say that word?" 4tLaxalt, who said he had not known Casey as long as the others, nevertheless credited him with saving President Reagan's presidential campaign from what he termed organizational "shambles" and "near bankruptcy."

"He marshaled that money like no one has ever seen in politics," the Nevada senator said, adding, "I abelieve that were it not for bill Casey, Ronald Reagan would not now be president." Laxalt said Casey's appointment to the CIA post was not a "political payoff" for Casey's rescue of the 1980 campaign.

In response to Goldwater's criticism that Casey should resign solely on the basis of his mistaken judgement on Hugel, Laxalt said Casey had told him that he would dispel the impression that Hugel's appointment was made "precipitously."

Casey "will indicate plainly to the committee that the Hugel appointment was a matter of intense consideration within the agency and a consensus was actually reached within the agency to go with Mr. Hugel. He had the requisites they felt they needed at the time," Laxalt said.

The three men also said that testimonial lunches for Casey have been organized for Aug. 3 and Aug. 6 in New York and washington.

Meanwhile, as the White House was operating on the assumption that the Senate Intelligence Committee investigation of Hugel and Casey would be completed this week, the senators were considering hiring Fred Thompson, veteran of the Watergate investigation, as special counsel for an extended inquiry.

The expectation within the administration is that the Senate probe will be wrapped up before Reagan goes to California for a vacation in early August. "It would be intolerable to have this situation drag out for weeks on end," a White House official said.

Just as friends have begun to rally around Casey, "there's also a sense of rallying around the chairman [Goldwater]," one Senate source said. "He really has the interests of the agency at heart."

Goldwater's call for Casey's resignation was followed by an assent from Republican whip, Sen. Ted Stevens (R-Alaska). Sen. Willian V. Roth Jr. (R-Del.) is so far the only Republican on the committee to join Goldwater in calling for Casey's resignation.

But Sen. David F. Durenberger (R-Minn.) was believed by some sources to be on the verge of urging Casey's resignation. Sen. Harrison H. Schmitt (R-N.M.) was also described as being "upset" with Casey for his failure to consult with the committee more often.