A personal note sent last week from Rep. Jack Kemp (R-N.Y.) to his colleague, Rep. Bill Gradison (R-Ohio), is a symptom of the deepening trouble the Republican Party faces over the budget deficits.

All last week, the headlines and television stories told of disagreements among Republican members of the Senate Budget Committee, frustrating plans of Chairman Pete Domenici (R-N.M.) to meet or exceed President Reagan's goal of holding net program spending in the 1986 budget to the 1985 level. Divisions within the GOP majority allowed the minority Democrats to pick Reagan's budget to pieces.

That was bad news for the administration, which counted on cohesion in Senate GOP ranks to produce some kind of a budget approaching Reagan's guidelines. But there is worse news waiting in the House, where the Kemp- Gradison exchange showed signs of a fundamental philosophical and political division that could force Reagan to come down from his safe perch and soil his hands in the hard work of budget construction.

Last Thursday, Kemp agreed with Gradison's suggestion that they cool the rhetoric and call off further exchanges of a public correspondence that had driven an increasingly sharp wedge into House Republican ranks. But their truce is temporary, and the fundamental differences between them -- and the two schools of Republican economics they represent -- will become obvious when the House Budget Committee, on which both serve, begins to mark up its version of the spending blueprint for 1986.

It began back on Jan. 3, when Kemp issued a "special report" to GOP members on the letterhead of the House Republican Conference, which he heads. Attacking those he said are framing the budget debate as a choice between "deep budget cuts, no matter how they may injure constituents . . . (and) sharp tax increases," Kemp said, "There is a way to cut through this Gordian knot: America can grow its way out of the deficit."

Kemp cited the nonpartisan Congressional Budget Office as confirming that a 5.2 percent annual growth in the economy and Treasury borrowing rates of 7 percent would "balance the budget by 1989. Is this economic pie in the sky?" he asked. "Not at all"; it can be realized easily with comprehensive tax reform to lower rates and -- that Kemp favorite -- "a modernized gold standard (that) would dramatically lower interest rates and provide liquidity to sustain the recovery."

That was more than Gradison could swallow. A brainy Harvard Business School grad and sixth-term member from Cincinnati, Gradison has inherited from retired Rep. Barber B. Conable Jr. (R-N.Y.) the role of spokesman for orthodox conservatism on the Budget and Ways and Means committees. In a Jan. 16 "Dear Republican Colleague" letter, Gradison said he was "surprised -- and disappointed -- by the use of the Conference as a forum to espouse so improbable a view (as Kemp's) as if it were House Republican policy. I have understood it to be Republican policy both in the White House and the Congress to act to reduce the deficit -- not to sit back and hope it will go away."

He said it was "simply untrue" that CBO's estimates supported Kemp's conclusion and said the report had distorted history "to support what is clearly an a priori conclusion." At bottom, Gradison said, Kemp was asking Republicans to "bet the country and the party" on a plan that boiled down to a plea to "gun the money supply."

Kemp replied on Feb. 13, charging that Gradison's letter was "unfair and disappointing" and asserting that the original report did not "say that deficits would be eliminated solely by economic growth." On Feb. 25, Gradison fired his last round, asserting that Kemp represented "a minority view among House Republicans," and adding that he still objected to the conference report "that ignores spending as a way to reduce the deficit and actually argues that we can grow our way out of the deficit."

This is much more than an academic argument between two bookish congressmen; it is a symptom of a serious split in the House GOP. Kemp and his allies, who wrote the economic plank of the 1984 Republican platform, are firmly opposed to tax increases, major defense reductions and domestic cuts that would hit important constituencies. Kemp has promised to produce a "jobs creation budget" of his own, which a spokesman said envisages only $30-$35 billion of spending reductions -- just two- thirds of what Reagan is seeking. Gradison and the conventional conservatives in the GOP will press for deeper cuts and fight Kemp's effort to move tax reform onto the agenda ahead of action on the deficits.

Reagan can ignore this Republican civil war, if he chooses, and follow his pattern of blaming the budget problems on congressional Democrats. But the din in the GOP is getting loud enough so the voters may notice.