"To the residents of the Gaza Strip, the Israeli army warns you against obeying the orders of the terrorist Hamas or having any contact with it," the recipient of one such pre-recorded message quoted it to AFP as saying.
"Know that Hamas is spending millions of dollars on tunnels used for hostile and terrorist acts against the state of Israel," it said. "This money should have gone to infrastructure, education and health projects."
The movement's Gaza interior ministry said that dozens of Gazans received the calls. The Israeli military did not immediately respond to a request for comment.
Israeli officials on Sunday said troops had uncovered a tunnel running from Gaza 450 metres (yards) into Israel and allegedly intended as a springboard for attacks.
Israeli daily Yediot Aharonot said Friday that Israel had known about the tunnel for over a year and let Hamas leaders continue to pour resources into it.
"Sometimes, as part of the war of minds, [Israel] lets them invest money, wrack their brains, waste human resources, and only towards the end of the digging, does it hold up a stop sign," the paper's defence correspondent Alex Fishman wrote.
Friday's phone blitz came on the anniversary of the October 18, 2011 release of Israeli soldier Gilad Shalit in exchange for 1,027 Palestinians held in Israeli prisons.
Shalit was snatched in June 2006 by a group of Hamas and other militants who snuck into Israel through a tunnel and took their prisoner back to Gaza the same way. Hamas feted the lopsided prisoner swap as a major victory.
"Resistance is our way to destroy the (Israeli) occupation and break its will and is our way to liberate our prisoners and our holy places," it said Friday in a statement to mark the anniversary.