* Conflict has killed more than 40,000
* PKK leader held on island prison
* PKK says urged to take its fighters out of Turkey
By Ozge Ozbilgin
ANKARA, Jan 4 (Reuters) - Turkish officials have made "important progress" in talks with jailed Kurdish militant leader Abdullah Ocalan to try to end a near three-decade insurgency by his supporters, a senior ruling party official said on Friday.
Prime Minister Tayyip Erdogan's chief adviser said on Monday Turkey had begun discussing disarmament with the Kurdistan Workers Party (PKK) militant group, and on Thursday two Kurdish lawmakers paid a rare visit to Ocalan in his island prison.
"Talks have reached a certain stage, some important progress has been made and some results have been achieved, or will be achieved," Nurettin Canikli, deputy chairman of the ruling AK Party's parliamentary group, told reporters in Ankara.
"The aim is to end terrorism, all efforts are being made for this," he said.
Talks with the PKK, which is designated a terrorist group by Turkey, the United States and the European Union, would have been unpalatable to Turkish public opinion only a few years ago.
Ocalan, who founded the organisation in 1974 to fight for an independent Kurdish state, is widely reviled by Turks who hold him responsible for the deaths of more than 40,000 people since the PKK took up arms in 1984.
Erdogan is under pressure to stem the violence - which has included bomb attacks in major cities as well as fighting in the mountainous southeast - particularly with presidential elections next year in which he is expected to stand.
Murat Karayilan, the acting PKK leader who is sought by Interpol, was quoted as cautiously welcoming the talks but said the government must show it was serious and allow PKK fighters to have direct contact with Ocalan.
"It would be a shame if the current process resulted in efforts to dissolve the PKK rather than aiming at resolving the Kurdish issue," he told Firat News, a website close to the PKK.
"For a solution, the Turkish side has to put its project on the table ... Nobody took up arms to have fun. There's a reason why these armed groups exist," he was quoted as saying.
Erdogan's government has widened cultural and language rights for Kurds, who make up about 20 percent of Turkey's 75 million people, since taking power a decade ago. But Kurdish politicians want more reforms including steps towards autonomy.
In July 2011, a month after his AK Party won a third term, Erdogan proposed peace talks with the PKK, and leaked recordings indicated senior intelligence officials had held secret meetings with PKK leaders in Oslo.
But the initiatives ran aground, and the last nine months have seen some of the conflict's bloodiest violence.
Pro-Kurdish Peace and Democracy Party (BDP) deputy Ayla Akat Ata and prominent Kurdish politician Ahmet Turk went on Thursday to Imrali, an island in the Marmara Sea where Ocalan has been in virtual isolation since his capture in 1999.
Few details of those or any previous talks have been made public.
"The Turkish government is not asking us to drop our guns, their demand has been to take our armed forces out of Turkey's borders," Karayilan was quoted as saying, adding that the group had been promised "safe passage" out of Turkey.
"Where are we expected to take them? To South Kurdistan? We have forces there already. But the Turkish side expects us to take the first steps, and how can we trust them," he said, referring to Iraq's autonomous Kurdistan region.
The PKK uses the remote Kandil mountains in northern Iraq as a base from which to stage attacks on Turkish territory.
With any hint of concessions to the PKK fiercely opposed by nationalists, it is not clear what the government believes it can offer to negotiate a truce.
The Radikal newspaper said on Friday one of the main issues discussed with Ocalan was a new Turkish constitution. Four political parties sit on a parliamentary commission set up after general elections in 2011 to forge a new constitution, which Erdogan has pledged will boost political freedoms.
Kurdish politicians have long demanded fairer representation and an end to discrimination in the nation's laws.