Oman's foreign minister spoke to his Israeli counterpart on Monday, Muscat said, the first contact since Israel normalised ties with the United Arab Emirates last week.
Yusuf bin Alawi subsequently spoke with a top Palestinian official, Oman added.
The Israel-UAE deal, announced by United States President Donald Trump on Thursday, is only the third such accord Israel has struck with an Arab country, and raises the prospect of similar deals with other pro-Western Gulf states.
Bin Alawi and Israel's Gabi Ashkenazi spoke via telephone about “recent developments in the region”, Oman's foreign ministry said on Twitter.
Bin Alawi told Ashkenazi that Oman “clearly reaffirms its position calling for a comprehensive, just and lasting peace” in the Middle East.
Bin Alawi also called for a “resumption of the peace process in order to satisfy the legitimate rights of the Palestinian people who aspire to an independent state".
While Oman and Israel do not have formal diplomatic relations, there have been several contacts between the two states, including in 2018, when the late sultan Qaboos received Israeli Prime Minister Benjamin Netanyahu in Muscat.
Also on Monday, bin Alawi spoke with senior Fatah official Jibril Rajub, who expressed his “appreciation of the role of the sultanate and its balanced and wise policy towards Arab issues and, foremost, the Palestinian question”, according to Oman's foreign ministry.
The Palestinian Authority has voiced its “strong rejection and condemnation” of the Israeli-Emirati deal.
Gulf Cooperation Countries (GCC) Bahrain and Oman have welcomed the deal, while Saudi Arabia, Kuwait and Qatar have yet to comment.
Home to Islam's holiest sites, Saudi Arabia would face sensitive political calculations before a formal recognition of Israel.