"We are getting to the negotiations with clean hands and great desire to get to an arrangement that will end the conflict," Moshe Ya'alon said.
Israeli Justice Minister Tzipi Livni lauded the prospect of peace in a statement.
"Four years of political stagnation are coming to an end," she said.
To understand the excitement over the rumored concessions -- if they are indeed a possibility -- one needs to go back to the dawn of the Six-Day War.
In June 1967, Egypt, Syria and Jordan massed their troops on Israel's borders in preparation for an all-out attack.
Israel launched a pre-emptive strike on June 5, targeting Arab airfields and destroying the Egyptian air force on the ground.
Egypt had set up a sea blockade against Israel the day before, according to the U.S. State Department Office of the Historian.
Military skirmishes and bellicose gestures between Israel and many of its Arab neighbors had reached a crescendo. Syria threatened to push Israel into the sea. Egypt allied with the Syrians.
Israel captured lands from its enemies and occupied them.
The United Nations demanded in Security Council Resolution 242 that Israeli troops withdraw and give the lands back. The resolution required all states involved to recognize each other's existence.
At first, Arab countries refused to recognize Israel as a state, and Israel retained the captured territory.
Over time, things changed. Egypt and Israel made peace in a deal brokered by U.S. President Jimmy Carter at Camp David in 1978, and Israel handed the largest captured chunk, the Sinai Peninsula, back to Egypt.
Jordan and Israel mended relations.
Israel withdrew from Gaza, bulldozing its former settlements on its way out.
But it has hung on to the West Bank, and the Palestinians have refused to recognize Israel as a state.
After decades of waiting, many hold out hope that new negotiations could ultimately bring about a resolution.