ANKARA/BEIRUT (Reuters) - President Tayyip Erdogan said on Sunday that Turkey will "clean" its entire border with Syria in a sign that the Turkish offensive on the Syrian Kurdish YPG group in northern Syria's Afrin region could be extended further.

Since Turkey's assault in Afrin began nine days ago, it has increased tensions between Ankara and the United States, which has supported the YPG in other parts of Syria in the fight against Islamic State.

"Step by step, we will clean our entire border," Erdogan said in a speech after the army said it had captured Jebel Bursaya, a hill that Turkish media had described as a critical position in recent days.

Erdogan said last week that Turkey might conduct operations eastwards inside Syria all the way to Iraq to clear the YPG from its frontier. Syrian Kurdish officials have pledged to meet any wider Turkish assault with an "appropriate response".

More than a week into their offensive, however, Turkey and allied Syrian rebel groups have made only modest progress fighting into hilly terrain against entrenched foes.

The U.S.-backed Syrian Democratic Forces, an alliance of militias of which the YPG is the strongest, had said earlier on Sunday that there was intense fighting in the area.

A war monitor, the Syrian Observatory for Human Rights, said Turkey and its allied rebel groups had taken the hill, which overlooks the major Syrian town of Azaz, which they also hold.

The Observatory said Turkish air strikes had killed three people in one family early on Sunday in Afrin. Turkish bombardment also damaged an ancient temple, it said.

(Reporting by Tuvan Gumrukcu; Editing by Raissa Kasolowsky)