Senior White House officials seriously discussed the possibility of asking budget director Bert Lance to resign as questions about his personal financial dealings mounted, White House press secretary Jody Powell said yesterday.

In the first such acknowledgement since the Lance controversy began, Powell said he and other senior aides discussed the possibility of a Lance resignation "at some length." They rejected such a course as an injustice to an innocent man, he said.

"It would have been easy for somebody from the White House to go over to the Executive Office Building [site of Lance's office] and say, 'Look, Bert, it's going to be tough . . . Why don't you just tell them to stick it?" Powell said.

But the press secretary added that "if we had cut and run on this man who had done nothing illegal or unethical . . . I don't know how we could have lived with ourselves."

Powell spoke to reporters at a breakfast meeting yesterday as the administration's defense of Lance continued - a defense, Powell said, that might not be "politically advantageous" to President Carter.

"Unfortunately," he said, "this is not a situation where what was right was also necessarily politically advantageous."

At the White House yesterday, Lance was a highly visible member of the administration. He attended a ceremony in the Rose Garden at which the President signed an executive order establishing a presidential management intern program under which the government will provide two-year federal internships in government agencies for up to 250 persons a year with advanced degrees in public management.

Lance later had lunch with Carter at the White House, and Powell said he would be surprised if the subject of Lance's financial dealings did not come up. The White House also announced five new government reorganization studies that have been launched under the direction of Lance and his staff at the Office of Management and Budget.

Powell, in his role as White House spokesman, has been the most active member of the administration in defending Lance. Both in public appearances and private conversation, he has repeatedly challenged reporters to produce evidence "of one thing that Bert Lance did that was illegal or unethical."

In an appearance on NBC-TV's "Today" show Wednesday morning Powell came as close as he had to criticizing the news coverage of the Lance affair.

He said one question put to the President at his news conference Tuesday - dealing with the competence of an OMB director who has overdrawn his own bank account since taking office - was "sort of [a] demogogic question."

In the same interview, Powell criticized Time magazine for questioning Lance's use of a corporate aircraft while director of OMB to attend an Associated Press meeting when officials of Time had offered use of their corporate aircraft to fly Lance to a meeting of the magazine's editors and advertisers.

"That is what I mean by people who set standards they can't . . . live up to," the press secretary said.

The seven bank overdrafts that Carter was questioned about Tuesday occurred between January and June of this year. Six of the overdrafts lasted for one day and the other for two days.

The seven overdrafts this year are not central to the Lance controversy. What the comptroller of the currency investigated were earlier overdrafts, some lasting for weeks and amounting to more than $100,000, that occurred in accounts Lance maintained in his own bank. The comptroller's report cleared Lance of any illegality or violation of banking regulations, but criticized some of his banking practices.

Yesterday, the White House released a telegram to Lance from Y. A. Henderson of the Calhoun First National Bank - the bank Lance once headed and where the overdrafts this year occurred - saying the bank was at fault.

In addition to Powell Lance has been quietly acting to shore up his position. Two days ago, for example, he hosted a lunch for eight columnists during which he vigorously defended his behavior as the head of two Georgia banks in response to skeptical questioning.

Despite the administration's strong defense of of Lance, Business Week magazine, an influential business and financial publication, yesterday called for him to resign as OMB director "because he has demonstrated that he dies not deserve public trust and confidence."

In the same editorial, the magazine said the President's support of his old friend was "a serious mistake."

According to knowledgable officials however, there has been no weakening in the determination of the President and his aides to stick with Lance, although they recognize the danger to Carter's credibility should new, damaging information concerning Lance be revealed.