Foes and backers of a plan to ease Yemen's president out of power fought each other with stones and clubs on Tuesday, deepening the country's chaos as Washington considered a request from the leader to fly to the United States.

Youth activists, who have led months of protests against President Ali Abdullah Saleh's 33-year rule, were split on whether he should leave the country - saying it might ease the conflict but could also let him escape justice.

Saleh bowed to months of protests and international pressure by agreeing last month to a deal that granted him immunity from prosecution over his violent crackdown on a popular uprising but saw him hand over power to his deputy.

Far from resolving the crisis, the settlement has caused further tension between groups who opposed the immunity deal, and those who backed it - many of whom have since joined an interim government.

Activists said at least 20 people were injured in the clashes in the capital, Sanaa, between supporters of the Islah party, which backed the immunity deal, and the Houthi movement, a Shi'ite rebel grouping in the north of the country.

Washington and Saudi Arabia, which borders Yemen, both fear continued chaos would allow al Qaeda to build on its already strong presence in the country, which is close to key oil shipping lanes.

After another bout of violence on Saturday - when protesters said Saleh's forces killed nine people who had joined a mass march against the immunity deal - the president vowed to give way to a successor and go to the United States.

U.S. State Department spokesman Mark Toner said on Tuesday Washington was still weighing Saleh's travel request.

Despite reports to the contrary, the United States is still considering President Saleh's request to enter the United States for the sole purpose of seeking medical treatment, Toner said. Only at the end of this internal review process will a final visa adjudication be made.

Anti-Saleh protesters said they were in two minds about the possible U.S. trip.

We are at a loss, between our desire to see Saleh go and avoid Yemen sliding into civil war, and the desire to see him tried for his crimes, said Samia al-Aghbari, a protest leader who was detained briefly after Saturday's violence.

If he (Saleh) is away and forbidden from being part of the political atmosphere in Yemen, it may help, I see the point of that. But he still has money and weapons in the country and if this doesn't change, nothing will change at any level in Yemen, said activist Hamza Shargabi.

OVERLAPPING CONFLICTS

Any suggestion that Saleh is taking up sanctuary in the United States would be highly controversial among activists and opposition figures who have accused Washington of backing Saleh as an ally in the campaign against al Qaeda.

He has this relation with the U.S., its war on terror, and torturing people in the name of that war, and putting people in prison, said Shargabi. Anything can happen in the name of the war on terror.

Hostility against the United States was fanned by Yemeni media reports that Washington's envoy in Sanaa had described Saturday's march as a provocative act, shortly before Saleh's forces cracked down on the protest.

In a statement on Monday, a group of protest organisers demanded Washington recall U.S. Ambassador Gerald Feierstein, whom they called an advocate and defender of Saleh's ruthless oppression of his people, almost from the start of his assignment in Yemen.

Al Masdar Online, one of the publications which attended a briefing with Feierstein, cited him as saying, in Arabic translation: Being peaceful isn't just about not carrying weapons. If 2,000 people decided to march on the White House, we wouldn't consider it peaceful and we wouldn't permit it.

The U.S. embassy in Sanaa did not respond to requests for comment on the remarks.

The top counter-terrorism official in Washington - which wages a campaign of drone strikes against alleged al Qaeda members in Yemen and assassinated Anwar al-Awlaki, a U.S. citizen, earlier this year - urged Saleh's deputy Abd-Rabbu Mansour Hadi on Sunday to show restraint with protests.

Any successor to Saleh would face overlapping conflicts including renewed separatist sentiment in the south, which fought a civil war with Saleh's north in 1994 after four turbulent years of formal union.

Islamist fighters have seized chunks of territory in the southern Abyan province. Fighting there has forced tens of thousands of people to flee, compounding a humanitarian crisis in a country where about half a million people are displaced.

(Reporting by Mohammed Ghobari; Additional reporting by Laura MacInnis in Honolulu; Writing by Joseph Logan; Editing by Matthew Jones and Christopher Wilson)