Israeli Agriculture Minister Uri Ariel gave Medvedev a tour of a research centre last week and reportedly asked him if he wanted the drone on display there as a gift after the Russian premier took an interest in it.
Footage broadcast on Israeli television shows Medvedev enjoying himself flying the drone with its remote control.
Medvedev accepted the gift -- but it was not clear whether Ariel was allowed to give it away.
The drone was made by Spanish company Alpha Unmanned Systems, while its thermal camera was from US company Flir, the Jerusalem Post reported.
There were also reports that Medvedev did not receive the remote control or the thermal camera with the drone.
Haaretz newspaper reported that staff at the Volcani Centre, which carries out agricultural research, had so far refused to give the components away.
Ariel's ministry issued a statement acknowledging that the drone was given to Medvedev, but said the gift had been approved by "the relevant professional officials".
The Movement for Quality Government in Israel called for Ariel to pay for the drone from his own pocket.
Opposition politicians slammed Ariel's largesse.
Nachman Shai of the opposition Zionist Union said the minister from the far-right Jewish Home had been duped into handing "knowledge and technologies to the wrong countries".
Stav Shaffir, a lawmaker also from the Zionist Union, called for Ariel's resignation.
The Volcani Centre and Israel's foreign ministry refused to comment.
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