"Recep Tayyip Erdogan expresses his deep regret over the incident and underlines the willingness to do everything possible to restore the traditionally friendly relations between Turkey and Russia," the Kremlin said in a statement.
In a message to Putin, Erdogan said Ankara has no desire to worsen its relationship with Moscow, which he sees as Turkey's "friend and strategic partner," and called for joint efforts to solve the crisis in the region and to fight terrorism, according to the statement.
Erdogan said Turkish authorities have launched an investigation into the death of the Russian pilot, adding that a Turkish citizen suspected of complicity in the case is under investigation, Xinhua reported.
The Turkish president also expressed his sympathy and condolences to the family of the deceased Russian pilot, Oleg Peshkov.
Earlier this month, Erdogan sent a letter to Putin, reiterating his wish to restore relations with Russia.
Relations between Russia and Turkey have soured after the latter downed the Russian bomber near the Turkish-Syrian border for alleged airspace violation, which Russia denied and considered as a hostile act.
Putin described the attack as a "stab in the back" and ordered a broad range of economic sanctions against Turkey.
Moscow has repeatedly said relations between the two countries could be restored only after Ankara apologizes and provides compensation to Russia for the downed aircraft and to Peshkov's family.