Russia, which has mooted holding such a Middle East conference in the past, is a member of the Quartet of Middle East negotiators, along with the European Union, the United States and the United Nations.
Moscow is the only Quartet member talking to Hamas, the group that controls Gaza but is snubbed by Israel and the West, and also has good contacts with Israel.
Mubarak said: "I affirmed Egypt's support for holding the proposed international conference in Moscow to push peace efforts, and our support for everything that contributes towards achieving a just and comprehensive peace in the region."
Egypt, like other Arab states, has called for a solution to the Palestinian-Israeli conflict based on two states and has dismissed Binyamin Netanyahu's, the Israeli prime minister, conditional proposal for a demilitarised Palestinian state.
"What is the basis for a solution to this problem? Those are international legal norms and certain principles including the principle of two states, discussions on [Israeli] settlements and the future capital," Medvedev said.
"These are difficult questions but Russia is ready to help solve them."
Egypt is the first stop on Medvedev's four-day Africa trip that also takes him to Nigeria, Namibia and Angola.
Cairo is Russia's leading trading partner in the Arab world, with an annual turnover of $4.1bn in 2008.
Egypt is the biggest consumer of Russian wheat exports, with more than four million tonnes estimated this season, and the issue topped the agenda in Tuesday's talks on trade between the two coutries.
Russia could set up long-term contracts to sell grain to Egypt once a quality control issue is resolved, a Russian official said.
Wheat shipments from Russia have been under scrutiny in Egypt since mid-May, when a prosecutor ordered a probe after dead bugs and impurities were found in Russian wheat imported by a private Egyptian firm.
The cargo was ordered to be sent back.
Rachid Mohamed Rachid, Egypt's trade minister, recently announced that Russia had suggested having a long-term agreement to export wheat to Egypt based on Egyptian specifications.