Mr Blair said that the Hamas movement, which holds power in the Palestinian enclave, is in contact with Egypt over the issue and that Cairo is prepared in principle to take action. All "responsible" players in the region should be working towards an immediate cessation of the hostilities which have now entered their 11th day, he said.
The former Prime Minister was speaking in Jerusalem after an intense night of fighting in Gaza as Israeli forces expanded their ground and air offensive.
Israel continued to ignore mounting international pressure for a ceasefire in the assault, which has claimed more than 500 lives, around a quarter of them civilians, according to the United Nations.
Israeli troops have fought pitched gun battles with Hamas militants as they targeted strategic buildings and smuggling tunnels linking the tiny enclave with Egypt.
Mr Blair, who yesterday spoke to Israeli Foreign Minister Tzipi Livni, told BBC Radio 4's Today programme: "There are circumstances in which we could get an immediate ceasefire, and that is what people want to see.
"These circumstances focus very much around clear action to cut off the supply of arms and money through the tunnels that go from Egypt into Gaza.
"I think if there were strong, clear, definitive action on that, that gives us the best context to give us an immediate ceasefire and start to change this situation.
"From my conversations, not just with Tzipi Livni but the Israeli Prime Minister and Defence Minister and others, I think that is the one basis on which we could bring a quick halt to this. Otherwise, I think we are in for a protracted campaign."
Mr Blair, who is the envoy of the Middle East Quartet of the UN, US, EU and Russia, said that talks on the issue were ongoing within the international community and between Israel and Egypt. And he said that, while he and other international representatives refuse to speak to Hamas, the movement's leadership in Gaza was well aware of the position from their own discussions with Egypt.
It was "difficult to judge" whether Hamas was ready to take the necessary steps to end the violence, said Mr Blair.
But he added: "I hope so, because if they truly do care about the people in Gaza, there is a possible way through this which would have an immediate halt and cessation of hostilities, and that is obviously what any responsible person should try to achieve...
"The immediate problem is not that people aren't talking to Hamas, because the Egyptians talk to Hamas, so it is not as though Hamas don't understand what the issues are or the position of the international community.
"The possibility of getting, within the next few days, a cessation of hostilities revolves around a package that is pretty clear to people."
He added: "I think the Egyptians, in principle, are prepared to do this. They want to do it. They recognise it is in their own interests as well.
"The question is, can this be put together in such a way that we get the immediate ceasefire that people want to see and then we have to address the longer-term question of how we get Palestinian unity."
Mr Blair said he had made representations to the Israeli authorities about access for humanitarian supplies to Gaza.
He said: "For anyone living in Gaza, it is hell, it is bound to be. You are in a situation where you are in an effective war zone.
"It is not a very large piece of territory, it is one of the most densely populated areas in the world. Hamas positions are well dug in actually inside the civilian population, so the notion that having a war going on around Gaza is going to be anything other than a humanitarian catastrophe is absurd, obviously."
Mr Blair urged incoming US President Barack Obama to engage with the Middle East peace process as soon as he is inaugurated on January 20.
"The most important thing for the new administration is to grip this, focus on it," he said.
"It is in my view absolutely central to the security not just of this part of the world, but all the world. We have got to grip it and sort it and if we do that with the requisite dedication and energy and commitment, we can resolve it."
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