Thousands of Gazans have poured into Egypt for a fourth day, despite Egyptian attempts to reseal the border.
For the first time many Palestinians used cars to cross, rather than going on foot.
Hamas, which rules Gaza, reacted to Israeli attempts to isolate the hostile territory with a blockade by blowing open its Egyptian border on Wednesday.
Hopes for talks between Hamas and its rival Fatah to tackle the crisis have been dealt a blow.
In a speech, President Mahmoud Abbas reiterated he would only talk to Hamas if it retreated from its June takeover of the Gaza Strip.
Hamas, which had indicated it would accept an invitation to talks, says this insistence effectively sabotages any prospect of negotiations.
This Gaza crisis began nine days ago, when Israel cut off the seaside territory, denying it supplies. It said it was aiming to subdue rocket fire from Gaza, but its actions were widely condemned.
Map of the Egypt-Gaza border area
'Let us live'
On Saturday, the governor of the Egyptian border province of North Sinai said that Egypt would continue allowing Palestinians to come across the border, and that it would help them stock up on supplies.
EGYPT-GAZA BORDER
He said the security forces had been instructed to facilitate the Palestinians' passage and that he was co-ordinating with ministers in Cairo to have fresh stocks of commodities sent to the border area.
The BBC's Heba Saleh, in Cairo, says it is clear that Egypt is now making a virtue of necessity.
Hundreds of thousands of people have surged into Egypt to buy supplies since the first breaches in the border wall were made.
Unconfirmed reports quoting the Egyptian foreign minister said that at least 38 Egyptian security personnel had been injured in incidents with Palestinians on the Gaza border.
But Cairo knows that any use of force against the Palestinians would come at a very high cost.
It would enflame domestic and Arab public opinion and severely undermine Egypt's credibility as a regional leader and peacemaker, our correspondent says.
Israel alarm
Israel, alarmed at the ongoing breakdown in security on the Egypt-Gaza border, has closed the main road running along the border. Tourism sites and hiking trails have been closed.
GAZA BLOCKADE
Security measures have been increased, according to the Israeli military, on fears that Israeli citizens could be vulnerable to attacks by Palestinians now free to travel in the area.
The breaching of the border has blown a big hole in the policy of trying to weaken Hamas by isolating it and sealing it in, says BBC analyst Sebastian Usher.
Some new system of controlling the border must now be found and it seems unlikely that this can be achieved without Hamas's involvement and agreement, he says.
Meanwhile, UK former PM Tony Blair - now a peace envoy to the Middle East - told the BBC the situation was "terrible" and a change in strategy was needed.
"I do think for the Gazan people and for the people in Gaza who are prepared to be part of this process, we need a different and better offer to make them"
Asked if that meant talks with Hamas, Mr Blair said: "Well I don't think you can give out as the sort of price of this communication with Hamas. Because having sent all these rocket attacks over into Israel and Israel retaliate, you then change your strategy.
"I don't think that's the right way to deal with it. But I do think for the Gazan people and for the people in Gaza who are prepared to be part of this process, we need a different and better offer to make them."
In his speech on Saturday, Mr Abbas reduced the prospect of talks with his Hamas rivals when he repeated demands that they reverse their takeover of Gaza, which he called a "crime".
Mr Abbas also urged Gazans to stop firing rockets into southern Israel, saying it gave Israel an "excuse" to punish the territory.
He said he would renew his offer to take control of Gaza's border crossings instead of Hamas during talks with the Israeli Prime Minister Ehud Olmert on Sunday.
The two are engaged in a new US-backed peace process that has excluded - and is rejected by - Hamas.
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