Facing budget constraints and increased public resistance to big spending, Secretary of Health, Education and Welfare Joseph A. Califano Jr. has been trying to prove since he took office that he is a hard-headed administrator who wants to wipe out fraud and waste.

The feisty Califano still favors some of the costly social programs that he helped initiate a decade ago as President Johnson's domestic policy chief.

But now, he wants to erase his big-spender image and show that money can be saved by tight administration without gutting the programs to help those who genuinely need it.

The new image he desires would improve his credibility on Capitol Hill by showing that he is not another wasteful welfare liberal. Associates claim it is not just a pose. "He really believes in it," said one official.

In his campaign, Califano has embraced the "fleeing fathers" (Nab-a-Dad) program. Designed to track down fathers who flee from child-support payments. It was a program at which liberals cringed when it was first suggested years ago by Senate Finance Committee Chairman Russell B. Long (D-La.). The aim of the program to recoup welfare costs.

Califano brought in Robert A. Derzon to streamline and cut costs on Medicare-Medicaid and then sacked Derzon last September for allegedly not moving fast enough replacing him with a young financial expert from Citibank Leonard D. Schaeffer.

Califano initiated a program to track down government employes who are collecting welfare in addition to their paychecks.

He ordered a report from his inspector general, Thomas D. Morris, alleging that there is up to $6.5 billion a year in waste, questionable outlays and outright fraud in HEW. The report led Congress to chop $1 billion out of his budget request last fall.

He initiated a tug-of-war with the states, still undecided, over the three-month waiting period before state and local governments must turn over Social Security taxes for their employes to HEW. If HEW received the money earlier, it could invest it in government bonds and draw up to $200 million a year in extra interest.

Califano has also given key policy slots to many of his former law partners and associates from the Johnson White House.

Stamford Ross, who attend Harvard law school with Califano and worked with Califano in the Johnson White Houe was recently confirmed as commissioner of Social Security in charge of about $100 billion a year in Social Security and welfare funds.

Ben W. Heineman Jr., former member of Califano's law firm (Williams & Connolly) and who has been Califano's chief executive assistant at HEW, is being named assistant secretary for planning, a post being vacated by Brookings Institution scholar Henry Aaron.

Frederick M. Bohen, a White House associate in Johnson years, who has been HEW executive secretary under Califano, is moving up to assistant secretary for management.

Whether Califano's campaign against waste will win him the full confidence of fiscal conservatives on Capitol Hill is unclear.

Some of his personnel changes have led to allengations by some that Califano, who hs a reputation for being combative and bossy, is trying to surround himself with these whose chief qualification is loyalty, not professional knowledge. Some believe he is in danger of cutting himself off from infusions of independent thought and judgment.

Others say the charge is a bum rap. "He doesn't want other people to make the decision, but he does listen to your arguments and if they make sense he'll go with you." said one HEW official.

One top Capitol Hill staffer said he doubts that Califano ever will persuade the fiscal conservatives that he is one of them. But the staffer said he sees nothing invidious about the appointment of former Califano associates and partners.

"Stan Ross is a crony but he is a brilliant guy, and Heineman is politically sensitive and probably will do well. I'm not an apologist for Califano but I don't see anything unusual about the appointments," the staffer said.