The Carter administration is putting the finishing touches on a billion-dollar program to deal with youth unemployment, likely to be the centerpeice of President Carter's domestic initiatives in his new budget.

Key administration planners say the program probably will involve about $800 million in start-up money in fiscal 1981, with room for between $1.3 billion and $2 billion a year in outlays after the plan is fully underway.

The effort, the result of a task force on the youth problem headed by Vice President Mondale, is designed to provide "compensatory" training to help disadvantaged youths make up what they missed, similar to Head Start for youngsters.

However, while the Head Start program only covers those in the elementary grades, the new effort would include junior high and high school youth, both those who remain in school and those who drop out.

The idea would be to provide youths with the job skills and work attitudes they need to get and hold jobs in today's labor markey. The effort would be run jointly by the new Education Department and the Labor Department.

Officials said that although the program would involve some consolidation of existing efforts, the bulk of the money would be channeled to local schools, which would operate the training classes.

Still undecided is precisely how large the program should be once it's fully in place. Officials said sponsors are calling for outlays of up to $2 billion or more, but White House budget-makers are pushing for a smaller package.

The new Carter initiative is based on the notion that unemployment among youths is concentrated among a relatively small group of disadvantaged teenagers and young adults who could be helped effectively by compensatory education.

White House studies and those of many private economists have shown the youth unemployment problem is most acute among blacks and other minorities, particularly in large cities. Current programs serve a much larger group.

The new youth program is expected to be one of the few -- and possibly the only -- sizable domestic initiative Carter will propose in his fiscal 1981 budget next month. The White House still is trying to hold down the budget.

Despite the prospect of a recession, officials now are planning a fiscal 1981 budget that provides for virtually no growth in total spending, after adjustment for inflation, with a deficit of about $15 billion -- half this year's size.

Total outlays likely will be about $600 billion, up from $547.6 billion this year, key sources said.

Moreover, to keep the deficit down in the face of a 3 percent or more "real" growth in defense spending, as Carter has promised Capitol Hill hawks, the administration will have to cut back on some domestic programs.

Current plans are to accomplish this by "targeting" some existing programs to specific groups that really need the aid, without slashing the big Democratic social programs such as welfare and education.