The sudden illiness of Premier Hedi Nouira, the man who has been running Tunisia for the aged President Habib Bourguiba and is his official heir apparent, has raised new concerns about the future of their pro-Western government.

Coming after an abortive revolt inspired by Libya last month, Nouira's hospitalization here underscored Tunisian security concerns especially in view of Libya's continued propaganda assaults on his government.

Habib Bourguiba Jr., son of the president for life, was hastily dispatched to deliver special messages to President Valery Giscard d'Estaing and President Carter. He met with Carter and Secretary of State Cyrus Vance today in Washington after seeing Giscard here Wednesday.

While the contents of Bourguiba's message were not disclosed, the United States opened discussions earlier this month about military aide to Tunisia, U.S. officials had said these talks are likely to result in the United States airlifting arms, specifically armored personnel carriers, to Tunisia in a show of support.

[During his meeting with the Tunisian president's son, Carter said: "We have observed with deep concern any threat to the independence and freedom of the people of your country." He also expressed U.S. thanks for Tunisian support "given us in times of crisis and common challenge."]

There has yet to be an official medical bulletin on Nouira's condition since he was rushed here from Tunis by medical evacuation plane Tuesday night. Informed sources say that he suffered a stroke. Tunisian officials admit that Nouira would most probably be sidelined for some time.

Even if Nouira is able to return to work eventually, the regime's image seems bound to suffer since Tunisia will appear as a country run, as one Western analyst said, "by two sick old men."

The elder Bourguiba, the father of Tunisian independence and formally designated as "the Supreme Warrior," is officially 76. He is generally reckoned to be several years older, perhaps as much as five. He is prone to bouts of severe crippling depression that last for months. These alternate with periods of whirlwind activity.

Yesterday, he took Nouira's place at the regular weekly Cabinet meeting. The Cabinet's key decision was to name a new interior minister, Driss Guiga, now the Tunisian ambassador to West Germany and a former director of internal security. That seemed to indicate the continuing concern about the fallout from the uprising in the southern town of Gafsa, last month.

It is increasingly clear that the Libyan-trained guerrillas in Gafsa got far more local support than the Tunis government has admitted. The Tunisian army was hunting down rebels and keeping a lid on Gafsa and other towns more than 100 miles away for weeks after the abortive revolt in the unemployment-ridden region.

The silver lining for the Bourguiba regime was the almost universal anti-Libyan patriotic reflex that prompted all the major opposition groups from the Communists to the Moslem fundamentalists to rally round in varying degrees. At the very least, they all denounced Libyan leader Muammar Qaddafi as the hand behind Gafsa.

Ironically, Nouira's illness already seems to be providing Qaddafi with a way to disarm the broad hostility he has aroused in Tunisia. In a statement to the Libyan News Agency, Qaddafi said that his confrontation with Tunisia could end "with the fall of Hedi Nouira, the symbol of that regime," provided there is a new government made up of opponents to the current single-party state. If not, he said, "I fear escalation in the confrontation being led by the Libyan revolutionary committees against the Tunisian regime."

Radio Tripoli, meanwhile, reported that Nouira had been shot in a general uprising. It was the latest in a series of wild Libyan claims about conditions in Tunisia since Gafsa.

Qaddafi also took the opportunity to say that his dispute with France, which Libya has been accuring of having reoccupied its former Tunisian protectorate, has been resolved thanks to France's conciliatory attitude. The new director of the French Foreign Ministry's North Africa section, Serge Boisdevaix, is currently in Tripoli talking with Libyan officials despite the government's open complicity in the wrecking of the French Embassy there in protest against French backing for Bourguiba.