On its face, the Reagan administration's last-ditch effort to jump-start a Middle East ''peace process'' looks hopeless, at best. At worst, it could be dangerous. Failure could further inflame Palestinian frustrations and lead to still bloodier outbreaks of stonings, firebombings, shootings and beatings in the Israeli-occupied territories.

So the reviews were not surprising when Secretary of State George Shultz presented a rough outline of what the administration has in mind to a meeting under the auspices of the Council on Foreign Relations at the State Department last week. The response of the assembled Middle East experts from government, big business and the American Jewish and Arab communities ranged from skeptical to scathing.

If not one feature, then another would be quite unacceptable to the Israelis, the Palestinians or neighboring Arabs, it was said. ''Insanity'' was what one participant called the whole idea of a lame-duck administration's trying to play peacemaker in a year when both the United States and Israel face national elections.

The proposition sounds even crazier when you consider what Shultz is apparently aiming to do. He would telescope into a matter of months what the Camp David accords of 1978 were supposed to make happen in stages over a matter of years. There was to be a five-year period of ''autonomy'' and limited self-rule for the inhabitants of the West Bank and Gaza and negotiations leading to a settlement on the final status of the Israeli-occupied land. Presumably this would put an end to the conflict that has triggered five Arab-Israeli wars and violence since 1948.

As is often the case in this town, there is a cynical explanation: it is that the Reagan administration, after seven years of ineffectual fumbling with the Palestinian problem and less than a year to go, now wants to make one last show of having tried hard. But as is also often the case, the straightforward explanation has the ring of truth. The particular nature and ferocity of the recent turbulence in the occupied territories, its demoralizing impact on Israel, the threat of its spreading to Jordan or other Arab states in the region -- all this has convinced the administration's top people, most notably Shultz, that doing nothing about it would be recklessly irresponsible.

Even more important, if the various parties to the conflict are deeply divided on how to solve it, they are increasingly united on one thing -- that, as Shultz puts it, ''the status quo in the peace process is not an option.'' For 10 years, the ''status quo'' has been outright Israeli suppression of Palestinian expression of political views and aspirations. Real or suspected PLO sympathizers have been jailed, deported, confined to their houses or their towns or otherwise harassed. Of 28 freely elected West Bank mayors, all but eight have been deposed.

The ill-disguised objective has been to foreclose the opportunity for the Palestinians that the Camp David formula took pains to leave open: a free choice in favor of an independent Palestinian state at some point down the road. Some analysts now argue that ''de facto annexation'' may already have advanced so far that it is irreversible.

But the uncontrollable, contagious spread of violence in the past two months is powerful evidence that the strategy of suppression is also unsustainable. Indeed, Israel's best (or worst) efforts to crush the Palestinian movement have only served to polarize the issue and compound the spontaneous combustion. That's the downside.

The upside is that serious Israelis can no longer persuasively deny the downside and that there is a genuine sense of urgency and widening anxiety not just in Jerusalem and Washington but in Arab capitals. Egypt's President Hosni Mubarak was just here, plugging a new peace approach. Veteran Middle East trouble-shooter Philip Habib has just returned from sounding out King Hussein, who deeply worries that the West Bank violence will spill over into Jordan, where Palestinians make up more than 60 percent of the population. The assistant secretary of state for the Middle East, Richard Murphy, is just back from touching bases in Syria, Saudi Arabia, Israel and Egypt. Shultz himself will be heading to the region for a week of talks with Israeli and Arab leaders immediately after he sounds out Soviet Foreign Minister Eduard Shevardnadze in Moscow later this month.

Israeli Prime Minister Yitzhak Shamir is coming to Washington next month. By that time, Shultz expects to have a clearer idea of the sorts of concessions it would take, all around, to reach an agreement. Only when he has some sense of the substance as well as the best procedure would he engage U.S. diplomacy in a heavy way. So this is no time for predictions. What can be said is that hardheaded reappraisals are under way in the right places and by the right people, not the least of them Shultz.

In the past he has been severely faulted for not wanting to get involved -- the more so if it meant leaning on Israel. But past critics detect a significant change. ''Shultz really has come around to realize,'' says one, ''that we are all on the road to disaster if we don't make a major effort to bring U.S. pressure to bear -- on both sides.''