The athletes were disqualified from their events, banned from participating in future Olympics and in some cases stripped of their medals following the International Olympic Committee's investigation of widespread doping in Sochi.
The president said that the scandals surrounding Russian sport had sparked heated debates within the country, and that this "creates an extremely hard environment for achieving results".
"Forgive us for not having been able to shield you from this", Putin said, adding that the scandal had contributed to "very hard conditions" for athletes.
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 Putin's spokesman Dmitry Peskov as saying.
"Certainly, we're very anxious about our sportsmen and women who were not able to participate in the Olympics".
"For us certain things are truly odd", Putin said in televised remarks. "We'd like to know the answer", he said.
I sincerely wish you success and victories..."We also expect that our colleagues in global sports organizations will do their best to see that these worldwide organizations, sports unions don't turn into departments of state agencies of certain countries, no matter how powerful and important these states would seem at first sight". Putin acknowledged Russia's "flaws and mistakes, a careless approach to things that are essential in modern sports", but also warned global sports bodies against becoming "departments of some powerful states".
There are "hundreds of thousands, millions, of fans who love you and hope that you win", the Russian leader said.