Jordan also made it clear that military maneuvers involving Iranian-backed forces could trigger a military confrontation between Iran and Israel.
Israeli Prime Minister Benjamin Netanyahu is also expected to meet with European leaders to discuss Iran next week.
"When you consider the advanced weapon systems - surface to surface missiles and anti aircraft systems - that the Iranians want to deploy in Syria, it becomes clear that they must be prevented from doing so in all of Syria and not only within a limited distance from the Israeli border", Chagai Tzuriel, director-general of the intelligence ministry, told Reuters.
Moscow says there is an agreement to hold a three-way meeting between Russian Federation, the United States and Jordan to discuss the de-escalation zone in southern Syria.
Russian Foreign Minister Sergey Lavrov issued a statement on Monday saying that only Syrian military forces should be present operating in the areas bordering neighboring Israel and Jordan.
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US State Department spokeswoman Heather Nauert warned on May 25 that the US would take "firm and appropriate" measures against Damascus in the event of President Bashar al-Assad ordering an offensive in southern Syria. A separate report in a Saudi newspaper said the regime had agreed that Iranian militias would stay 25 kilometres away. The Iranians denied they had fired any rockets.
The unconfirmed report said the negotiations were conducted last weekend between Iran's ambassador to Jordan, who was in one hotel room in Amman with Iranian security personnel, while in the next room there were senior Israeli security officials, including the deputy head of the Mossad.
Sources briefed on the trip told Israel's Channel 10 reporter Barak Ravid that the two officials will be ratifying an existing agreement between Israel and the USA on how to deal with Iran.
According to Elaph, Iran may have been willing to negotiate with Israel this weekend as a result of losses Tehran has sustained in recent Israeli attacks in Syria.