* Army steps up campaign against al Qaeda in south Yemen
* Says Afghans, Somalis, Chechens among 37 militants killed
* Sharp rise in militant attacks on state targets in south (Adds militants saying abducted soldiers released)
By Mohammed Mukhashaf
ADEN, Yemen, May 4 (Reuters) - Yemen said at least 37 al Qaeda militants were killed in the southern province of Shabwa on Sunday as the army intensified an offensive to root out foreign and local Islamist fighters holed up in some of the country's most impenetrable areas.
The report by state news agency Saba came shortly after a military source told Reuters a suicide bomber killed six soldiers by their outpost in Shabwa, one of two provinces where the army has been fighting al Qaeda and its allies.
The source, speaking from the battlefront, also told Reuters that 20 soldiers were wounded in the bombing.
The number of attacks against the army and security forces in the south has risen sharply since Yemen's U.S.-backed army launched its anti-al Qaeda offensive last week.
Western countries fear further destabilisation in Yemen, which also faces separatists in the south and unrest in the north, could give more space to al Qaeda in the Arabian Peninsula (AQAP), the local branch of the global Islamist militant movement, to plot attacks on international targets.
Washington has used drones for several years to kill members and leaders of the group. In April, air strikes - presumed to have been carried out chiefly by U.S. drones - were launched on central and southern provinces of Yemen, which Yemeni authorities said killed some 65 al Qaeda militants.
Saba quoted a military source as saying that most of the militants killed in Sunday's fighting were Saudis, Afghans, Somalis and Chechens, as well as other nationalities.
The source also said the army destroyed a number of vehicles and weapons belonging to the militants.
President Abd-Rabbu Mansour Hadi said in a speech last week that 70 percent of al Qaeda fighters in Yemen were foreigners, although AQAP later denied this.
Separately the defence ministry website said security forces had destroyed a vehicle carrying several "terrorists", killing two of them and wounding four in Shabwa.
"BROKEN THE BACKBONE"
Late on Saturday, suspected militants in Shabwa threw hand-grenades at a central bank building, wounding two soldiers, a local official said.
AQAP and its local ally, Ansar al-Sharia, have been waging an insurgency in southern Yemen for more than three years, battling both government forces and local tribal militias.
Hundreds of people have died in bombings, suicide attacks and commando-style raids by al Qaeda against military and government facilities and foreign nationals.
A Twitter account by the name of Ansar al-Sharia news said the group had released eight Yemeni soldiers whom they had taken hostage last week in Shabwa, after tribal mediation.
It was not immediately possible to verify the authenticity of the Twitter account but a tribal source in Shabwa confirmed the news to Reuters.
AQAP's main base has been the mountainous al-Mahfad area in Abyan where militants fled in 2012 after the army, with the help of U.S. drones, ousted them from towns they had seized during the chaotic uprising against then-president Ali Abdullah Saleh.
Yemeni Defence Minister Mohammed Nasser Ahmed was quoted by Kuwaiti newspaper Al-Seyassah saying the army's offensive in Shabwa and Abyan had "broken the backbone of the al Qaeda terrorist group and destroyed a number of its strongholds, equipment and weapons". (Additional reporting by Mohammed Ghobari in Sanaa; Writing by Yara Bayoumy; Editing by Sami Aboudi and Jon Boyle)