He added the military had established a call centre to answer questions from individuals and organisations about the mobilisation.
Amateur footage posted on social media since Putin made the announcement has purported to show hundreds of Russian citizens across the country responding to military summons.
In the Siberian city of Yakutia, footage showed men with their faces obscured embracing family members -- some in tears -- before boarding a bus.
Popular Telegram channel Mash distributed footage showing long lines of people waiting to board a plane in the eastern Khabarovsk region as part of the mobilisation drive.
A video purportedly filmed in the southern republic of Chechnya showed dozens of young men at a local recruitment office flanked by police.
The footage could not be independently verified by AFP.
The defence ministry has not distributed official video from recruitment centres and has not given information about the number of people handed summons.
Putin on Wednesday announced a “partial mobilisation” of reservists and defence minister Sergei Shoigu said the drive would ultimately see 300,000 people called up for service.
More than 1,300 people were arrested across Russia on Wednesday during improvised demonstrations against the mobilisation.