Peres said: "The emir has asked that Israel recognise the Hamas government as the expression of the Palestinians' will and said Israel should be pragmatic and start negotiations [with Hamas]."
He said direct talks would continue to be impossible as long as Hamas refuses to abide by accords signed between Israel and Palestine and while the organisation continues to deny Israel's right to exist.
"We will not hold talks until they [Hamas] change their attitude and [as long as they] refuse to recognise Israel."
Peres declined to respond to questions about whether Qatar was helping to win the release of an Israeli soldier held by Palestinian groups.
He is the first major Israeli official to travel to any Gulf Arab country since he visited Qatar in 1996 as prime minister.
The six Gulf Arab countries, all US allies, do not recognise Israel.
However, Qatar maintains low-level ties with the Jewish state and hosts an Israeli trade commission.
Peres, who is on a two-day visit to the country, arrived on Monday to participate in the the Doha Debates, a discussion in which the Israeli deputy leader answered questions from an audience of 300 Arab students.