The government will increase efforts to track down "runaway" fathers and force them to support their children, Health, Education and Welfare Secretary Joseph A. Califano Jr. said yesterday.

The federal-state program already brings in nearly $900 million from absent parents - almost all fathers - but Califano said he hopes to collect almost $600 million more though a series of new actions.

On other issues, Califano told a news conference he preferred a modified version of the administration's stalled welfare bill to a compromise fiscal relief bill now being considered in Congress. He also announced that Joseph Bartlett, a Boston attorney and former undersecretary of commerce, will be his principal adviser "on whether to recommend Social Security coverage for all federal, state local government and non-profit organization employes.

With strong backing from Massachusetts Gov. Michael Dukakis, an official of the National Governors' Association, Califano said he is initiating "Project Responsibility," a series of actions designed to help the states track down more runaway fathers and get the money from them.

It includes creation of a new National Institute for Child Support Enforcement to help train officials in establishing paternity, tracing fathers and using legal techniques. In addition, HEW will create a Child Support Enforcement Reference Center as an information clearinghouse and develop a model computer system to help in parent-location efforts and in billing absent parents.

Califano said that the current program collected $423 million in fiscal 1977 in child-support payments from absent fathers of children living on welfare. In addition, he said, it collected $460 million from absent fathers of children who weren't on welfare but whose mothers or guardians applied for assistance in tracking down the father and making him pay. Of the total collected, $275 million went for administrative costs, so the net yield was about $600 million, officials said.

Califano said the goal of Project Responsibility is to raise the total of child-support payments from fathers of welfare children from $423 million a year to $1 billion in fiscal 1979. Aides said the department also hoped to step up collections from absent fathers of non-welfare children, but Califano didn't announce any specific goal.

The "runaway fathers" program was mandated by Congress four years ago on the initiative of Sen. Russell B. Long (D-La.). It requires every state to set up an agency to help locate such fathers and force them to support their children. The federal government pays 75 percent of the costs and helps in tracing them through Social Security, tax and employment records.