Turkey's President Recep Tayyip Erodgan on Monday apologised over the downing of a Russian jet last year, the Kremlin said, suggesting the possibility of rapprochement between Ankara and Moscow.

Erdogan's office, however, said that he had expressed "deep regret" for the incident, but had stopped short of officially apologising.

The Russian Sukhoi jet was shot down by a Turkish fighter plane near the Syria-Turkey border in November 2015, causing a major diplomatic rift between the two countries.

Turkey has opened an investigation into a Turkish citizen who is accused of killing the pilot after the plane came down in Syrian territory, a spokesperson for the Kremlin said.

Details of the case into the Turkish suspect - whose actions in shooting at a pilot after he ejected from his aircraft would have broken the Geneva Conventions - will be kept secret for "national security reasons", Turkish media reported on Monday morning.

"The head of the Turkish state in his message expressed his sympathy and deepest condolences to the family of the dead Russian pilot and said sorry," Kremlin spokesman Dmitry Peskov told reporters.

Presidential press spokesperson Ibrahim Kalin confirmed in a statement that Erdogan had offered condolences to the pilot's family, and said the premier was keen for both countries to take steps "without delay" to improve Turkish-Russian relations.

"We are pleased to announce that Turkey and Russia have agreed to take necessary steps without delay to improve bilateral relations," the statement said.

The downing of the Russian fighter jet - which Turkey said violated its airspace during a bombing raid over Syria - led Putin to impose trade restrictions on Ankara, saying they would not be lifted unless Erdogan apologised.

The Kremlin spokesperson said Erdogan wants to do "everything possible for the restoration of the traditionally friendly relations between Turkey and Russia."

"Erdogan has announced that his country is prepared to work with Russia to work against the crisis in the region and fight terrorism."

In the wake of the announcements, Turkish Foreign Minister Mevlut Cavusoglu said he would be attending a Black Sea Economic Cooperation meeting in the Russian town of Sochi on 1 July.

Russia and Turkey have been at loggerheads for months - Moscow's President Putin is a key financial and military backer of Syrian President Bashar al-Assad, while Ankara plays a prominent role in supporting the country's opposition rebels.

Erdogan's office did not officially confirm or deny his reported statements, but was expected to release a statement on Monday afternoon.

Russia and Turkey have traditionally enjoyed good relations, with a strong trading relationship and an average of over three million Russian tourists visiting Turkey annually.

However, the numbers of Russian tourists visiting Turkey fell sharply at the end of last year as a result of the jet incident, as well as of economic woes in Russia.