Tebboune and Putin discussed "bilateral relations in their call between the two countries, especially energy cooperation", the Algerian presidency said in a statement.
Algeria has had warm ties with Moscow for decades, but Africa's biggest gas exporter has also become crucial for Europe's energy supplies in the fallout of Russia's invasion of Ukraine.
Algeria is a major buyer of Russian arms, and in 2021 bilateral trade was worth three billion dollars, despite the coronavirus pandemic.
Tebboune is also set to pay a state visit to former colonial ruler France in May, but officials have not specified which country he will visit first.
Algeria, which pumps gas directly to Spain and Italy via undersea pipelines, has in recent months hosted a string of top European officials -- including French President Emmanuel Macron in August -- seeking to find alternatives to Russian energy supplies.
The North African country is in a decades-long struggle with its regional rival Morocco, particularly over the disputed Western Sahara territory, and cut off all ties with its neighbor in 2021 over alleged "hostile acts", which Rabat has denied.
As a permanent member of the United Nations Security Council, Russia has a direct influence over the Western Sahara situation.