President Reagan, facing one of the most critical personnel decisions of his term, must choose a new Environmental Protection Agency chief who will restore public confidence in the agency, solve a political problem for the White House and get Congress off his back--all without changing his administration's basic environmental policies.

If that sounds like a big order, the president's supporters and critics want the new administrator to be widely known and recognized in the environmental field, to know his or her way around Washington and Capitol Hill and not to be identified with industry.

"It's extremely important that just the right person be selected, so we can't rush it," presidential personnel director Helene Von Damm was quoted as saying last week. "We have a group of people there we feel can hold down the fort, so we feel no need to rush things."

But the sensitivity of the job suggests that more than caution is behind the measured pace of the White House deliberations. There may be no sure candidate.

Congressional aides say that any Reagan candidate's chances of confirmation are slim if the nominee agrees with Reagan's policies, which he has shown no inclination to change. Instead, Reagan has lashed out at the "environmental extremists" who have attacked them.

"The president must realize that he's misjudged the public's concern," said Bob Harris, a Washington consultant and Carter administration appointee to the Council on Environmental Quality. "I don't believe the president has learned any lesson. He probably sees this as a political hassle. I have some skepticism that the person going in would have any sense of change in attitude."

That is a concern to Republicans, who have begun to fear that the White House is handing Democrats a potent political issue. But it is not clear if their concern is strong enough to override the president's policies through congressional action.

Some Reagan supporters continue to hope that a new administrator can help calm the storm at the EPA and get the agency back in the public's good graces.

"You need somebody who can convey the impression that the administration cares a lot about the environment," a senior EPA official said. "People who watch the network news have to feel that the administrator cares about the environment."

Names of potential successors began to surface even before administrator Anne M. Burford resigned last Wednesday, saying she had to leave because she had "become the issue."

The difficult question for the White House is how to choose a replacement who will not, under the glare of the nomination and confirmation process, become "the issue" again.

Hours after the news of Burford's departure, environmentalists and leading members of Congress were listing qualifications for her successor.

Sen. Robert T. Stafford (R-Vt.), chairman of the Senate Environment and Public Works Committee, said he was thinking along the lines of "a Russell Train or a William Ruckelshaus." But the Sierra Club called for someone with "no ties to industry."

Presumably, that leaves out Ruckelshaus, the first EPA administrator under President Nixon. He is now an executive with Weyerhaeuser Co. and has pressed for changes in water and air pollution laws to put more emphasis on the costs of compliance to industry.

Stafford has rejected John R. Quarles, the first name floated by the administration, largely because of his industry ties. Quarles, a former deputy EPA administrator, is now an industry lobbyist and has worked to weaken provisions of the Clean Air Act.

Train, president of the World Wildlife Fund, might be acceptably clean of industry ties, but he has attacked the administration for turning the EPA into a "paper tiger" with policies that "will destroy the agency as an effective institution for many years to come."

Such words are not likely to endear him to Reagan, who last Friday said his critics would not be satisfied "until the White House looks like a bird's nest."

Reagan said much the same thing in November, 1980, before his election. Environmentalists took his renewed attacks as a clear signal that he will not be amenable to a candidate with a strong environmental background, as many members of Congress of both parties have suggested.

Last Friday, nine House Republicans wrote to Reagan, suggesting that he appoint "someone who has a strong record of experience and interest in environmental protection."

There is a large pool of potential candidates with solid Republican credentials but little administrative or environmental experience, a disadvantage in handling an agency as complex as the EPA.

"If they want to turn the agency around, they've got to get somebody with experience so they can hit the ground running, rather than spend six months getting . . . kicked around ," a Republican Senate aide said.

Meanwhile, critics and supporters of the administration said there are good candidates acceptable to all sides but only if they are given the resources to do the job. That involves a policy change, they said.

Interior Secretary James G. Watt, who has had the greatest influence on the administration's environmental policy, told the New Yorker magazine in May, 1981, "We will use the budget system to be the excuse to make major policy decisions."

When he said that, Burford was working out of an office in the Interior Department, formulating policies that would guide the EPA. By August, she had devised a budget proposal to cut the agency's budget by 25 percent.

"Anyone who takes that job and doesn't deal with that problem is guaranteed to fail and the agency with them," said William Drayton, a Carter administration EPA official and outspoken critic of Reagan's policies. "The question is whether there is anyone who has the qualifications of the Archangel Gabriel and Peter Drucker," who is considered the guru of modern business management.