The Kremlin leader also told athletes who are due to compete in Pyeongchang next month, as independent competitors, that he was sorry officials had failed to protect them from the fallout.
"Forgive us for not having been able to shield you from this," Putin said during a reception at his residence outside Moscow on Wednesday, adding that the scandal had contributed to "very difficult conditions" for athletes ahead of the Games.
"You should focus on the result and on your victory," he added. "Nothing should distract you from this."
"I wish you luck," Putin said, adding he wanted them to be "courageous".
He said the exclusion of many Russian sports men and women over doping was "strange", reiterating his pledge to support those who have been prevented from taking part.
The International Olympic Committee banned Russia from the 2018 Winter Olympics after a report on a vast, Moscow-backed doping scheme in previous Games.
Only 169 of Russia's athletes have been allowed to take part in the Games, under a neutral flag as an "Olympic Athlete from Russia".
Speaking to reporters, Putin's spokesman Dmitry Peskov said the president had tasked ministers with organising an alternative competition for those banned from the 2018 Games.
Putin "reminded the cabinet of the decision to organise a competition for those of our athletes who had not been allowed to take part in the Olympic Games," Russian news agencies quoted Peskov as saying.
The Kremlin spokesman added that athletes who win at the alternative event would receive compensation equal to what Moscow normally awards Olympic medalists.
Pro-Kremlin newspaper Izvestiya reported this week that the alternative competition for banned athletes would take place in the Black Sea resort of Sochi which hosted the 2014 Winter Olympic Games.
Speaking to the athletes, Putin reiterated his complaint that the ban against Russia was politically motivated.
"For us certain things are truly strange," he said.
He said he hoped that international sporting organisations would not become annexes of "certain countries", a thinly-veiled jab at the United States.
Last year, Putin claimed United States was behind Russian sports doping allegations to influence March presidential elections widely expected to extend the Russian leader's Kremlin term until 2024.
Putin has consistently denied the Russian state has run a vast doping scheme, instead blaming individual sporting officials.
'Greatest sporting nation'
The athletes thanked Putin for his support, with hockey player Pavel Datsyuk presenting the Kremlin chief with a jersey featuring the slogan "Russia in my heart" and players' signatures.
Luger Roman Repilov for his part said the Russian athletes should emerge winners at Pyeongchang to show the world that "Russia is the greatest sporting nation."
"We should prove to them all that we are Russians, that we are strong, that those who are banning our teammates will not be able to break us," Repilov said.
"We should pull ourselves together and go there and win."
Snowborder Alena Zavarzina said the exclusion of other Russians was utterly unfair.
"I believe that what happened with our colleagues who cannot go to these Games is an egregious injustice," she said.
"We will fight and do everything in our power to bring home the medals that we deserve."
The 23rd Winter Olympics will open in Pyeongchang, South Korea, on February 9, featuring 102 events in 15 sports and athletes from all over the world.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)
Get Breaking news, live coverage, and Latest News from India and around the world on NDTV.com. Catch all the Live TV action on NDTV 24x7 and NDTV India. Like us on Facebook or follow us on Twitter and Instagram for latest news and live news updates.