Mr Raisi was killed in a helicopter crash in a mountainous area of north-western Iran, along with Foreign Minister Hossein Amir-Abdollahian.
State media confirmed they were killed after the helicopter they were travelling in came down on Sunday.
Ayatollah Khamenei said he offered his condolences “to the dear people of Iran”.
Mr Raisi, 63, had been tipped as a potential successor to the supreme leader.
State media said elections would be held on 28 June to choose a new president.
In the meantime, Vice-President Mohammad Mokhber has been appointed to assume interim duties.
Iran’s cabinet has also appointed deputy Foreign Minister Ali Bagheri Kani as acting foreign minister.
It was initially reported that the helicopter carrying the men and their entourage had made a hard landing in foggy conditions.
The area where the aircraft came down is close to the border with Azerbaijan, where Mr Raisi had been meeting President Ilham Aliyev.
According to local media he was there to open the Qiz Qalasi and Khodaafarin dams.
On Monday, the Iranian Red Crescent confirmed the bodies of the president and others who died in the crash had been recovered and search operations had ended.
“We are in the process of transferring the bodies of the martyrs to Tabriz [in Iran’s north-west],” the organisation’s chief said on state TV.
Iranian news outlet Tasnim, which is affiliated to the country’s Islamic Revolution Guards Corps, reported that President Ebrahim Raisi’s funeral will be held on Tuesday in Tabriz.
Mr Raisi, 63, was a hard-line cleric and his election as president in 2021 consolidated the control of conservatives over every part of the Islamic Republic.
In a statement following his death, the Iranian government said it would continue to operate “without disruption”.
