Chad's civil war is fomenting serious new tensions between the United States and France, with the two governments becoming increasingly suspicious of each other's motives and policy aims.

The Reagan administration has grown impatient with France's refusal so far to send fighter aircraft and combat troops into battle to crush rebel forces and wreck any ambitions that their patron, Libyan leader Muammar Qaddafi, might harbor about building a revolutionary axis across the Sahara.

The government of French President Francois Mitterrand, in turn, seems exasperated by what it perceives as Washington's obsession that Qaddafi is acting as a Soviet surrogate bent on seizing Chad as a crucial domino in the East-West struggle.

President Reagan's assertion Thursday that the United States has no intention of intervening militarily in Chad because that country lies in France's "sphere of influence" stirred controversy here because it seemed that Reagan was urging France to police former colonies in Africa as its primary role in a power contest with the Soviet Union and its allies.

Not only has Mitterrand taken pains to neglect France's traditional role as the military gendarme that imposes stability in former African colonies, but he also has firmly opposed the notion that developing countries should be perceived as pawns in a global chess match between East and West.

Some French commentators suspect that Reagan's reference to spheres of influence underscored a desire to confront Mitterrand with the exigencies of a security zone in Africa so that he might better comprehend the military measures that the U.S. administration deems necessary in Central America.

Under Mitterrand, France has become one of Western Europe's harsher critics of the Reagan administration's hard-line approach to Latin American leftists.

Mitterrand and French Foreign Minister Claude Cheysson share the belief of many European social democrats that much of the revolutionary turbulence in less-developed regions derives more from social injustice and economic exploitation than from machinations by Moscow and its allies.

French dismay at U.S. efforts to ascribe troubles in Central America to intensified East-West rivalry has turned to pique over Reagan's apparent determination to elevate the latest phase of Chad's long civil war to another test of superpower wills.

"The government would do well to reflect before it acts," warned Michel Jobert, who served in the Mitterrand government as external trade minister before resigning. "It is neither in France's interest nor its doctrine to proceed with head bowed into the great game proposed by the Russians and Americans."

Jobert, as well as several Socialist and Communist members of the government, denounced the pressure emanating from Washington through the black African leaders of Zaire, the Ivory Coast and Senegal, who have urged France to take a stronger military stance in the conflict in Chad.

Mitterrand's Socialist Party issued a strong endorsement of the government's cautious tack yesterday, saying that "France has better things to do than play gendarme in Africa again, even if the military threat seems to be a way of containing the armed intervention of Libya in Chad and forcing it to accept a negotiated solution."

The party declaration added that "the French government must resist the aggressive impatience of Reagan diplomacy."

French officials have complained that the Reagan administration has not shown sufficient understanding of the dilemma posed not only by Mitterrand's desire to brake French military involvement in Africa, but also by the need to maintain relations with all parties in the conflict.

Until the latest eruption in Chad, France was seeking better relations with the Qaddafi regime because Libya was viewed as an important trading partner that might help alleviate some of France's economic difficulties.

In addition, the French saw their efforts to cultivate channels with Libya, as with all African states, as a basic diplomatic strategy to restrain U.S. and Soviet influence on the continent and enhance their own importance as a third power dominant in North and West Africa.

Socialist-led France has strongly supported the United States in its defense buildup and nuclear arms policy. Mitterrand also gave the order to expel 47 alleged Soviet spies earlier this year in a move that soured French-Soviet relations.

But the current outlook in French-American relations is for the bad to get worse. The weakened French economy is receiving no help from high U.S. interest rates and the strong dollar.

And in Chad and Central America, the process of finding a lasting solution is likely to be long, arduous and fraught with further disagreement between Paris and Washington.