(Adds Libya seeking U.N. probe into NATO bombings)
* Gaddafi PM asks for international probe into NATO 'abuses'
* Tunisian army clashes with Libyan fighters
* Oil chief doesn't return in week's 3rd apparent defection
By Tarek Amara and Ulf Laessing
TUNIS/ZAWIYAH, Libya, Aug 20 (Reuters) - Libyan rebels battled for towns on either side of the besieged capital Tripoli on Saturday, and fighting spilled across the border into Tunisia where Libyan infiltrators clashed with Tunisian troops.
This week's rebel advances on Tripoli -- Muammar Gaddafi's last major stronghold -- have transformed the war by cutting the capital off from its main road link to the outside world and putting unprecedented pressure on Gaddafi's 41-year rule.
Washington says the veteran leader's days are now numbered, and reports have emerged of more defections from Gaddafi's ranks.
The six-month-old war came close to the Tunisian frontier after rebels suddenly seized the coastal city of Zawiyah just 50 km (30 miles) west of Tripoli, surrounding the heavily fortified capital and severing its vital supply routes.
In Tunisia, security sources said their forces had intercepted Libyan men in vehicles with weapons and fought them through the night in the desert. They reported several casualties, but did not say whether the fighters were Libyan rebels or pro-Gaddafi soldiers cut off from Tripoli.
Residents of the southern Tunisian desert town of Douz told Reuters by telephone that helicopters were swooping overhead and troops had been summoned from nearby towns to subdue the infiltrators, who rode in vehicles without number plates.
The imposition of a siege around Tripoli has trapped its residents and cut it off from fuel and food supplies. The International Organisation for Migration said on Friday it would organise a rescue operation to evacuate thousands of foreign workers, probably by sea.
Intense fighting continued in Zawiyah, home to an important oil refinery, on Saturday and rebels occupying the center of the city said pro-Gaddafi forces showed no sign of retreat.
"Gaddafi will try to take back Zawiyah at any price. He will keep shelling the hospital," said a rebel fighter as he prepared for midday prayers in the mosque of Bir Hawisa, a nearby village where many civilians are sheltering.
"We will not let that happen. We will fight."
East of Tripoli, fighting has been bloodier and rebel advances far slower. On Friday, opposition forces fought street battles in the city of Zlitan but suffered heavy casualties, a Reuters reporter said. A rebel spokesman said 32 rebel fighters were killed and 150 wounded.
NATO warplanes have hammered Gaddafi military targets since March under a U.N. mandate to protect civilians. Gaddafi's government has said the bombs have killed scores of innocent people, including 27 during a raid on Tripoli this week.
On Saturday, Libyan Prime Minister Al Baghdadi Ali Al-Mahmoudi spoke to U.N. General Secretary Ban Ki-moon by telephone requesting an investigation into alleged abuses by NATO, Libyan state news agency JANA reported.
JANA said Ban had promised to study the proposal.
In another potential blow to Gaddafi, a Tunisian source said Libya's top oil official, Omran Abukraa, had arrived in Tunisia after deciding not to return to Tripoli from a trip to Italy.
If confirmed, it would be the third apparent defection of a senior Gaddafi associate this week. A senior security official arrived in Rome on Monday, and rebels said on Friday that Gaddafi's estranged former deputy Abdel Salam Jalloud had joined their side in the western mountains.
The siege of Tripoli and the prospect of a battle for the capital have added urgency to the question of Gaddafi's fate. The leader has repeatedly vowed never to leave the country and rebels say they will not stop fighting until he is gone.
A senior U.S. official said on Saturday that the opposition must prepare to take over power soon. The United States is among more than 30 nations that have recognised the rebels' National Transitional Council (NTC) as Libya's legitimate authority.
"It is clear that the situation is moving against Gaddafi," U.S. assistant Secretary of State Jeffrey Feltman told a news conference after meeting Libyan rebel leaders at their headquarters in Libya's eastern city of Benghazi.
"The opposition continues to make substantial gains on the ground while his forces grow weaker." (Additional reporting by Missy Ryan in Tripoli and Robert Birsel in Benghazi, Libya; Souhail Karam in Rabat, Morroco; Writing by Richard Valdmanis and Peter Graff; Editing by Maria Golovnina)