The remarks appeared to be an effort by the Western-backed Abbas to turn the tables on Israel, which has cast him as responsible for the diplomatic deadlock and the surge of bloodshed.
Speaking to Channel 2, Abbas gave rare details on his domestic security drives, a touchy matter as many Palestinians deem such moves collaboration with their enemy.
"Our security forces go into the schools to search pupils' bags and see if they have knives. You don't know this," he said.
"In one school, we found 70 boys and girls who were carrying knives. We took the knives and spoke to them and said: 'This is a mistake. We do not want you to kill and be killed. We want you to live, and for the other side to live as well.'"
Abbas's administration and Israel coordinate security in the West Bank despite the stalling two years ago of US-sponsored negotiations on Palestinian statehood.
Netanyahu says he is open to renewing talks and that Abbas has been avoiding these while inciting violence with his rhetoric against Israel.
But Abbas told Channel 2 that the onus was on Netanyahu.
"I will meet with him, at any time. And I suggested, by the way, for him to meet," the Palestinian leader said in English.
Asked what became of that overture, Abbas said: "No, no—it's a secret. He can tell you about it."
Netanyahu's office had no immediate response.
Abbas' Palestinian Authority exercises limited self-rule in the West Bank under 1993 interim peace deals. Israeli forces now freely operate in PA areas, something Abbas described as sapping his credibility at home. He said he was willing to take action against Palestinians that Israeli intelligence deems a threat.
"If he (Netanyahu) gives me responsibility and tells me that he believes in (the) two-state solution and we sit around the table to talk about (the) two-state solution, this will give my people hope, and nobody dares to go and stab or shoot or do anything here or there," Abbas said.