Senate Republican leaders yesterday launched a week-long campaign to nail down GOP support for their deficit-reduction compromise with President Reagan, and White House chief of staff Donald T. Regan said there appear to be enough votes to pass it in the Senate.

After the first in a series of private huddles with GOP senators, Senate Majority Leader Robert J. Dole (R-Kan.) said the initial reaction was "pretty positive," although concern was expressed that the package might be picked apart on the Senate floor.

Democrats are expected to force politically distasteful votes on key items, including a proposal to halve the cost-of-living increase for Social Security recipients over the next three years. Some Republicans have said they fear that the package could unravel in the process.

But Regan, emerging from a strategy session with Dole, expressed cautious optimism. Asked how many votes the package will get, he said, "I think enough, but we'll have to see."

The compromise, negotiated earlier this month after the administration balked at a Senate Budget Committee plan, would cut annual deficits by more than half to less than $100 billion by fiscal 1988, largely by big reductions in domestic programs.

Reagan's proposed defense buildup for next year would be limited to 3 percent after inflation, a split-the-difference compromise between the administration's request for 6 percent and the Budget Committee's zero-growth proposal.

Despite the expected painful votes when the Senate takes up the package next Monday, members of Congress received good news as they trickled back from their 10-day Easter recess: Deficit projections are down slightly from earlier in the year.

In its mid-April update of projections in Reagan's February budget, the Office of Management and Budget said the deficit for this year is projected to be $213.3 billion, down $8.9 billion.

The fiscal 1986 projection was down $2.6 billion, which would marginally affect projected deficits of about $175 billion for next year under both Reagan's budget and the compromise.

The OMB attributed the change to a growth in national income stemming from anticipated increases in inflation; to increased tax collections, largely from higher corporate profits, and to a reduction in anticipated spending for programs ranging from farm credit to Social Security.