The president's budget threatens their unity.

The nation's governors are meeting here under unusually strained conditions. They face a major challenge from proposed federal fiscal policies, and serious threats to their own hard- won unity.

As the National Governors Association holds its annual winter session from today through Tuesday, the preoccupation will be President Reagan's budget and tax proposals. The Reagan budget calls for a 19 percent reduction next year in authorizations for federal aid to state and local governments -- the sharpest cutback in modern history and the steepest in any category of federal spending.

At the same time, the president has renewed his call for elimination of deductibility of state and local taxes as part of the tax-revision measure the Senate is to consider, a move that will inhibit the state's own revenue-raising efforts.

While Reagan is unlikely to get all his requested spending cuts, and the House has already rejected the repeal of deductibility, state and local government officials recognize that they are peculiarly vulnerable to this year's budget pressures.

That is why the potential divisions among the governors are particularly awkward this year. The obvious split is political, but the division on economic lines may be even more important.

This is the big year for gubernatorial elections, with 36 of them on the ballot. Twenty-seven of those 36 governorships are now held by the Democrats. And in 13 of the 27 states, the incumbent Democrats will not be running for reelection.

That spells large-scale political opportunity for the Republicans, who are now outnumbered 34 to 16 in governorships and are determined to move closer to parity this year. For a decade, under governors of both parties, the governors' association has attempted to create a bipartisan state presence and lobbying pressure in Washington.

But that cooperation faces serious strains. Last summer, at the meeting in Boise, the election of officers was delayed by a vehement Democratic protest of a fund-raising letter the Republican governors' association had sent out. It accused the Democrats as a group of being high-tax, big-spending liberals.

Tennessee Gov. Lamar Alexander (R), the 1986 NGA chairman, is respected and trusted by his colleagues. But Alexander dissents personally from the general opposition to repeal of deductibility, and he is actively promoting the election of more Republican governors and the 1988 presidential candidacy of former Sen. Howard H. Baker Jr. He will have to walk a careful line as governors' association chairman.

The situation is made vastly more complicated by the growing recognition of the two-tier economy of the United States and the growing polarization of states and communities between haves and have-nots.

The glowing national reports of declining unemployment, steady economic growth and low inflation mask the fact that important sectors of America -- agriculture, mining, energy and some lines of manufacturing -- are operating at recession or depression levels.

The disparities are evident in the fiscal condition of the states. While overall state finances appear to be in good shape -- with about $6 billion in surpluses and "rainy-day" funds projected -- almost one-third of the states were showing signs of serious economic pressure even before the new federal cuts hit.

A survey this month by the National Association of State Budget Officers found 14 states have been forced to cut their fiscal 1986 budgets since they were passed last year, in order to avoid a crisis. With a couple of exceptions, those states are not traditional big spenders, so the cuts are coming from muscle and bone -- not from fat.

They are states that are dependent on agriculture, mining and energy and such sick industries as textiles: Colorado, Hawaii, Idaho, Montana and Utah in the West; Arkansas, Mississippi, Oklahoma and South Carolina in the South; Iowa, Minnesota, Nebraska and Wisconsin in the Midwest. Vermont is the only northeastern state on the list, indicating how out-of-date the still-popular "Frostbelt"-"Sunbelt" dichotomy has become. New England and the Middle Atlantic states are probably the healthiest economies in the nation today.

The differences among the states carry an important policy warning to the administration and Congress: the domestic spending cutbacks that are coming will hit some communities much harder than others, unless some thought is given to cushioning the impact.

One such approach is embodied in the "targeted fiscal assistance" bill sponsored by Sen. Dave Durenberger (R-Minn.) and Rep. Bob McEwen (R- Ohio). With general revenue-sharing to local governments almost certainly ticketed for extinction this year, they are proposing to take half the amount of no-strings aid (or $2.3 billion) and pump it into communities with especially heavy tax burdens, for low-income populations and large needs. In New Jersey, for example, it would eliminate wealthy Princeton from unrestricted federal aid but put more money into hard-pressed Newark.

Such a program would indirectly help the struggling states as well. But lobbying it through Congress will require a unified effort from governors who are today anything but unified.