President Reagan has ordered federal departments to come up with another $10 billion to $15 billion in budget cuts to offset rising estimates in next year's deficit, the White House announced yesterday.

At the same time, House Budget Committee Chairman William H. Gray III released a new analysis of the fiscal 1988 budget deficit that predicts an $11 billion increase in the deficit over what was projected early this year.

Reagan ordered the new budget cuts as White House officials continued to discuss strategy before next week's deadline for an increase in the national debt ceiling. Spokesman Marlin Fitzwater rejected efforts by Senate leaders to relax the targets in the Gramm-Rudman-Hollings budget law as part of a compromise.

"We believe the target should be kept at the levels they are and that the president should have the authority that allows him to make spending decisions that will meet them," Fitzwater said.

Fitzwater said Reagan is seeking the new cuts because the economy has grown more slowly than expected and there have been spending "additions" in the first six months of the year.

The letter seeking reductions was sent to the Pentagon. However, since the Democratic-controlled Congress also has sliced Reagan's proposed defense budget, officials said the Defense Department would not be asked to come up with additional cuts.

The Gramm-Rudman targets call for a 1988 deficit of $108 billion. Reagan has claimed the budget he submitted earlier this year would meet that target; congressional analysts said Reagan's budget as submitted was $27 billion over the target. Congress rejected Reagan's budget and approved one including a $19 billion tax increase, which Reagan vowed again yesterday to veto.

The House Budget Committee's study, based on economic information provided by the nonpartisan Congressional Budget Office, projects that without changes in current spending or tax policies, the deficit for the fiscal year beginning Oct. 1 would be $182 billion.

If Congress and the president agree on legislation necessary to implement the $1 trillion budget passed last month -- including the tax increase that Reagan has said he will veto -- the fiscal 1988 deficit would still be about $150 billion, according to the analysis. That is some $42 billion higher than the Gramm-Rudman target.

The new analysis does contain a silver lining, however, projecting that the expected deficit for the current fiscal year will be $162 billion, $14 billion lower than earlier forecast. The decrease results from higher-than-expected revenues produced by the tax-overhaul law passed last year.

However, the study also predicts that slower economic growth, increased inflation and higher interest rates will continue to push the deficit higher than earlier projections after next year. It projects a fiscal 1989 deficit of $198 billion, $34 billion more than the last congressional estimate.

Both the House and Senate next week will take up measures that would ease the Gramm-Rudman deficit targets by substituting for the set yearly goals a requirement that Congress and the administration achieve annual deficit reductions of about $36 billion.