Kremlin spokesman Dmitry Peskov told reporters the pilots were "guided by co-ordinates" from "Turkish partners".
"There should not have been Turkish troops at those co-ordinates," he said.
But the Turkish military insisted the troops had been at the building for 10 days and that Russia had been advised of their position on Wednesday.
The two countries, which support opposing sides in Syria's almost six-year civil war, are working together to drive the Islamic State group from its last stronghold in Aleppo province.
Turkish-backed rebel fighters have been besieging al-Bab from the north since December, while Russian-backed government forces are advancing from the south.
Russian President Vladimir Putin was quick to call his Turkish counterpart, Recep Tayyip Erdogan, to offer condolences after Thursday's air strike, which also wounded 11 Turkish soldiers.
On Friday morning, Mr Putin's spokesman told reporters in Moscow that the causes of the incident were "clear".
"Unfortunately, when carrying out strikes against terrorists, our military were guided by co-ordinates that had been handed over by Turkish partners," Mr Peskov said.
Turkey's Deputy Prime Minister, Numan Kurtulmus, meanwhile said the air strike was still being investigated.
"According to initial information we received, it is a total accident," he was quoted as saying by the state-run Anadolu news agency. "But how it happened and how the co-ordination was miscalculated will be clarified."
The Turkish military subsequently issued its statement challenging Mr Peskov's account and stressing that the soldiers' position had been communicated the previous day to both officers at Russia's Hmeymim airbase in Syria and the Russian military attache in Ankara.
The air strike came as rebel fighters clashed with pro-government forces near al-Bab for the first time since coming within firing distance of each other.
Russia had to intervene to prevent further fighting at a village south-west of the town again on Friday, according to the Reuters news agency.
Both sides view al-Bab as a prize, and it is not clear whether they intend to fight for control of it.
The rebels have already entered the town's western outskirts, but pro-government sources told the New York Times on Wednesday that Russia and Turkey had agreed last month that government forces would enter the city, giving them access to a key water facility that supplies the city of Aleppo.
Meanwhile, the UK-based Syrian Observatory for Human Rights reported that at least 38 civilians in al-Bab had been killed by air strikes and artillery fire by Turkish-led forces in the past three days.
Turkey's military has denied that it mistakenly told Russian warplanes to bomb a building in Syria on Thursday, killing three Turkish soldiers.