GAZA, Jan 27 (Reuters) - The Islamist Hamas group in control of the Gaza Strip signalled a new readiness on Wednesday to resume Egyptian-sponsored reconciliation talks with the rival Fatah movement headed by Palestinian President Mahmoud Abbas.

Egypt has become openly frustrated with what it views as Hamas's stubborn resistance to a pact that would mend the worst rift in decades in the Palestinian independence movement.

Senior Hamas leader Mahmoud al-Zahar told a Gaza news conference that if Egypt were to guarantee implementation of what we agreed, we would be willing, to sign an agreement under which new Palestinian elections could be held in June.

Zahar's remarks on the long stalled Egyptian-backed efforts to heal a rift between the two main Palestinian political groups, were seen partly as an effort to ease tensions with Cairo over the killing of an Egyptian soldier in clashes at the Gaza border this month.

Hamas had previously refused to sign an Egyptian-drafted agreement between the Islamists and the secular Fatah movement, insisting it should be permitted first to propose its own amendments. Egypt rejected any additions to its document.

Abbas's Fatah group has already signed the document which calls for steps to resolve a dispute with Hamas that erupted in open warfare in 2007 when the Islamists seized control of coastal Gaza, and for holding a parliamentary and a presidential poll in five months' time.

An agreement could end a rift that has effectively divided control over the Palestinian territories, and limited the authority wielded by the Western-backed Abbas to the realm of the Israeli-occupied West Bank.

Zahar said Hamas wanted an Egyptian guarantee to make sure there would be fair elections.

We have fears that if things are not clear and if there are no guarantees from Egypt some of the parties may slip away from the implementation, Zahar added, alluding to Fatah.

Zahar said Hamas has nearly completed an investigation into protests in which an Egyptian soldier died on Jan. 6 when Hamas supporters protested an underground barrier Egypt is constructing along its border with Gaza to bar arms smuggling.

Hamas officials have avoided travel to Egypt, which is Gaza's main outlet to the outside world due to an Israeli blockade of its crossings with the territory, since that time. Sources close to Egypt said Cairo officials balked at attempts by Hamas officials to contact them indicating the tense relations. (Writing by Allyn Fisher-Ilan; Editing by Charles Dick)