Binyamin Ben-Eliezer of Israel's Labor Party, a former Israeli defense minister who often served as a government envoy to Mubarak, said he spoke with Mubarak just hours before the Egyptian president's speech late Thursday in which he transferred some authorities to his deputy but refused to step down.
This refusal angered hundreds of thousands of Egyptians demanding he relinquish his three-decade grip on power.
Anti-government demonstrations have rocked Egypt for weeks and have brought the country to a virtual standstill.
Describing his conversation with Mubarak to Israel's Army Radio, Ben-Eliezer said: "He knew that this was it, that this was the end of the road."
"He was looking for only one thing - give me an honorable way out. Let me leave in an honorable fashion," Ben-Eliezer said.
Another former Israeli defense minister, Shaul Mofaz, now an opposition lawmaker who chairs an influential parliamentary defense committee, echoed Ben-Eliezer's statement, telling the same station that Mubarak "wants to end it on his feet and not on his knees."
Egypt signed a peace treaty with neighboring Israel in 1979.
Israel is concerned that the disappearance of Mubarak, a longtime ally, could mean a breakdown of order in Egypt or the rise of the Muslim Brotherhood, Egypt's largest and best-organized opposition group, which is hostile to Israel. Both scenarios would threaten Israel's security.
In 2007, the Islamic group Hamas - which identifies itself as the Palestinian branch of the Muslim Brotherhood and is backed by Iran - seized control of the Gaza Strip. Last month, the Lebanese militant group Hezbollah, also backed by Iran, toppled the government in Lebanon, showing its growing influence over Israel's northern neighbor.
"At this very moment in our area the balance of power is changing, and the situation is working against Israel," Mofaz said. "Iranian radicalism is becoming stronger and improving its position."