Russia's athletics team and Paralympics squad were banned from last year's Olympic Games in Rio de Janeiro.
Russia's athletics team and Paralympics squad were banned from last year's Olympic Games in Rio de Janeiro after the World Anti-Doping Agency (WADA) found evidence of state-sponsored doping in the country.
Coe said Putin's comments on Wednesday represented "a large step in the right direction" towards getting Russian athletes readmitted to international competitions.
"You have to be encouraged by that observation and it's very important," Coe told BBC Radio 4 on Thursday.
"There's no doubt at all that the new Russian Athletics Federation is populated by people who I genuinely think have grasped the enormity of this challenge.
"That's a good statement to make and it's important we maintain that progress."
While Putin again denied accusations of state-sponsored doping, he acknowledged the previous anti-doping system had failed, which he said was "our fault and we should acknowledge it".
"I feel it's moving in the right direction," Coe said.
"The recognition that the system has failed the athletes and it's a system that clearly needs to change is something that we should be comforted by.
"Those observations today have taken us a little further down the road and that's a good thing."
Asked if Russia might be allowed to return to competition this year, Coe said: "I hope that there is a real possibility of that.
"The IAAF's objective is to make sure the (Russian) federation is fit for purpose, has the right people in and meets the criteria. So it does look as though things are beginning to come together."
WADA president Craig Reedie called Putin's comments a "sign of progress from the highest political levels".