Egyptian officials seeking to broker a deal between Palestinian groups have met Hamas leaders in the Gaza Strip.
Tens of thousands of Hamas supporters rallied on Friday
The move follows days of rising tensions between Hamas, the largest faction, and the Fatah group headed by Palestinian leader Mahmoud Abbas.
Mr Abbas is expected to use a speech to the Palestinian parliament on Saturday to threaten early elections - an option already condemned by Hamas.
Hamas has blamed Fatah for a shooting that targeted PM Ismail Haniya.
Mr Haniya's car was attacked by gunmen while crossing into Gaza from Egypt on Thursday. One bodyguard died in the incident.
A Hamas spokesman blamed a senior Fatah official, former security chief Mohammad Dahlan, for the attack.
The shoot-out at the Rafah border crossing led to more clashes on Friday, both in the West Bank and in Gaza City.
Some 32 people were injured when Palestinian police loyal to Fatah fought Hamas supporters in the West Bank town of Ramallah.
The power struggle between Hamas and Fatah is marked by huge ideological, political and personal differences between the leaders of both groups, the BBC's Alan Johnston reports from Gaza.
There have been several bouts of street fighting since Hamas won a sweeping victory in parliamentary elections last January.
Egyptian mediation has helped calm the situation several times.
But there is a sense the current tensions are on a higher plane than has been seen previously, our correspondent says.
Several efforts to form a national unity government have broken down.
With the attempt on Mr Haniya's life and Hamas' accusation that Mr Dahlan was involved, the leaders of the rival groups have now been drawn into the line of fire, our correspondent adds.
Hamas said it would boycott Mr Abbas' speech to the Palestinian legislature on Saturday in protest at "dangerous and bloody" recent events.
Senior Fatah official Saeb Erekat told the Associated Press that Hamas' decision was a "dangerous escalation".
US Secretary of State Condoleezza Rice told Reuters news agency she would seek tens of millions of dollars in extra funding to boost Mr Abbas' security forces.
Despite the tensions, Mr Haniya called for calm and unity among Palestinians at a rally marking the 19th anniversary of the founding of Hamas.
Masked gunmen took to the streets as tens of thousands of supporters at a Gaza City stadium heard Mr Haniya vow to pursue those who attacked his convoy and bring them to justice.
He said Hamas knew who was responsible, but gave no names.
Earlier, an official Hamas spokesman said the attack was organised by Dahlan, a key Fatah figure in Gaza and fierce critic of Hamas.
Mr Dahlan rejected the Hamas accusations, saying the governing party was trying to "mask its failures".
A Fatah spokesman said the attack was a "grave threat" to Palestinian unity.
The crisis in the Palestinian Authority has grown since Hamas won elections in January.
Western donors have withheld aid payments in protest at Hamas' refusal to renounce violence or recognise Israel.