(Adds talks on Guantanamo, paragraphs 11-16)

By Sue Pleming

ALGIERS, Sept 6 (Reuters) - U.S. Secretary of State Condoleezza Rice pushed for closer north African counter-terrorism cooperation on Saturday on a tour of an oil-rich region periodically hit by violence blamed on al Qaeda.

Rice, visiting Algeria and Tunisia a day after holding a historic meeting with Libyan leader Muammar Gaddafi in Tripoli, told reporters she had been saddened by recent attacks in regional giant Algeria that killed scores of people.

"The cooperation here is good," she said after talks with Algerian President Abdelaziz Bouteflika, adding the two countries had affirmed their "strong ties" in the discussions.

"But there is always more that you can do to tighten sharing of information, to make sure you have all the right channels to give technical support in terms of the terrorism threat," she said, referring to north African states generally.

"It is not a secret that it is a problem that has really been very salient in the Maghreb."

An estimated 125 people were killed in an upsurge of political violence in Algeria in August in attacks by al Qaeda's north Africa wing and clashes between the army and militants.

The periodic violence worries many in the region because the big country on the southern shores of the Mediterranean is a major energy exporter to Europe and its 34 million people are still searching for stability after years of political strife.

An Islamist revolt began in 1992 in Algeria after the then military-backed authorities, fearing an Iran-style revolution, scrapped an election an Islamist party was set to win. More than 150,000 people have been killed in the ensuing violence.

Rice said she also discussed security with President Zine al Abidine Ben Ali on a visit to Tunisia earlier in the day.

POLITICAL REFORM

Security concerns focus on oil- and gas-exporting Algeria, although pro-Western Morocco and Tunisia have also been hit by attacks by al Qaeda-aligned militants in recent years.

Rice said she had raised the question of Algerians held at the Guantanamo detention camp with Bouteflika and added the two countries had a good working relationship on the question.

"It is important to have both security assurances and human rights protection and we feel we have done well with Algeria on both," she said, reiterating the U.S. aim to close the centre.

Human Rights Watch this week urged Rice to press Algeria to ensure the well-being and status of Guantanamo detainees returned to Algeria recently. The group says four have returned to Algeria in the last two months while 21 remain in the camp.

Algerian officials have said any return of Algerians from the camp could only happen "without any condition or constraint".

U.S. officials say some governments will not take custody of their citizens held at Guantanamo, others would not treat their citizens humanely and still others are not willing to provide security guarantees Washington believes are necessary.

Hundreds of Algerian Islamist fighters joined the Afghan opposition to Soviet occupation in the 1980s and became involved with what is now al Qaeda in the 1990s.

Rice also said she had pushed for deeper political reform and free elections on her visit to Tunisia, long accused by critics of rights abuses and stifling press freedom.

On Friday Rice made the first trip by a U.S. secretary of state to Libya in 55 years, a move intended to end decades of enmity following the signing of a compensation package to cover legal claims involving victims of U.S. and Libyan bombings. (Reporting by Sue Pleming, Writing by William Maclean, editing by Janet Lawrence)

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