The Russian president said he wishes for a successor who is capable of leading a "wonderful country like Russia."
Medvedev penned an op-ed in a Russian paper that was highly critical of the U.S. president.
The former Russian president said he is pessimistic a Biden presidency would mean better ties between Washington and the Kremlin.
Russia's entire government—including prime minister Dmitry Medvedev—resigned abruptly on Wednesday, after Putin addressed the nation.
The U.S. has already announced incoming sanctions against sensitive electronic imports and senators want fresh restrictions on Russian finance and investments.
NATO and Georgian troops are currently conducting military training exercises.
While the president's personal approval is only a few points from its record peak, most Russians disapprove of the prime minister, the government and parliament.
"Donald Trump was sitting practically next to me," Dmitry Medvedev boasted.
The two nations' premiers shared their thoughts on linking payment systems, as Moscow said that no single currency should dominate global trade.
A former president, Moscow's mayor and Putin's onetime bodyguard top the list to succeed the current leader.
Dmitry Medvedev, constitutionally Russia's second most powerful official, accused Trump of giving power to Congress "in the most humiliating way."
Alexey Navalny summoned thousands in March and Moscow police issued a warning on the eve of the next protest.
Russia's top anti-corruption blogger claimed that the PM has a hidden empire of luxury assets.
Dozens of protesters were detained after thousands took to the streets to demand the resignation of the prime minister.
Analysts warn that recent government actions do not look supportive of women's rights.
"We should not delude ourselves," Medvedev said.
It seems he's a fan of a certain British detective.
The president has played a part in boosting United Russia's approval amid poll drop.
The prime minister suggested teachers should change careers if they wanted better pay.
Reserves are running too low to expect additional investment, he said.
The Kremlin is concerned about losing revenue from oil.
War of words between leaders grows increasingly bitter as political fallout continues
Facing an uncertain future, Russia's opposition leaders mourn the 'irreplaceable guy with brass balls'.