The US administration has hailed a call by Palestinian leader Mahmoud Abbas for fresh elections, expressing hope it could help end violence in the region.
Fighting between rival Palestinian groups has risen in recent weeks
But the Islamic group Hamas, which won the latest Palestinian elections, rejected the move as a "coup attempt".
Mr Abbas's call sparked more clashes between Hamas and his Fatah movement.
Overnight, a man was killed when gunmen attacked a Gaza training camp used by the presidential guard and on Sunday a Hamas minister's car came under fire.
Gunmen attacked a convoy carrying the Palestinian Foreign Minister Mahmoud Zahar, a Hamas member, security sources said, although there were no immediate reports of injuries.
Several gunmen were injured in exchanges of fires between the two rival groups across the Gaza Strip on Saturday.
Meanwhile, Palestinian election officials say they need at least three months to prepare for new presidential and parliamentary polls, the Assoicated Press reports.
They were speaking after meeting Mr Abbas to discuss his call for early elections.
"We explained to him the phases of holding elections, and how much time it would take for holding such elections, which is at least three months, including challenges," the head of the Central Election Commission, Hanna Nasser, said.
In a major policy speech in Ramallah on Saturday, Mr Abbas blamed Hamas for the crisis, saying the Palestinian people were suffering from an economic siege that had halved incomes.
The group's refusal to recognise Israel and renounce violence has triggered the suspension of Western aid to the Palestinians.
Escalating tensions in recent days between Hamas and Fatah have raised fears of civil war.
The US administration welcomed the call for fresh elections.
"While the elections are an internal matter, we hope this helps bring the violence to an end," White House spokeswoman Jeanie Mamo said.
Meanwhile, UK Prime Minister Tony Blair - who is touring the Middle East tour - urged the international community to back Mr Abbas's effort to break the deadlock.
Israel said Mr Abbas had its support in his bid to bring peace to the Palestinian territories.
"We hope that he will be able to assert his authority over all of the Palestinian people," government spokeswoman Miri Eisin said.
Many Palestinians say the president has no right to dissolve the government, which was elected in January and is due to remain in office until 2010.
Correspondents say it is not clear when or how new elections would be held.
It will be up to the Central Election Commission to try to find a legal way of carrying out Mr Abbas's orders, says the BBC's Nick Thorpe in Ramallah.
The Hamas government reacted angrily to Mr Abbas's speech, calling it "a coup against Palestinian legitimacy and the will of the Palestinian people".
Mr Abbas' call was welcomed by outside powers
Ahmed Yousef, an adviser to Palestinian Prime Minister Ismail Haniya, said the call for elections was a recipe for violence.
"I think this will lead to bloodshed because this is something against the constitution."
Several Palestinian factions based in the Syrian capital, Damascus, also rejected the call for early elections.
The past week has been marked by attacks, counter-attacks and mutual accusations.
Hamas blamed Fatah for a shooting that targeted PM Ismail Haniya on Thursday, but Mr Abbas, in his speech, denied there had been any conspiracy to kill Mr Haniya.