This claim is discussed by the German newspaper Die Zeit (in English) Vera Putina’s Lost Son:
There are those who are convinced that the Russian president spent the first nine years of his life with a family whose existence he continues to dispute to this day. They also believe he spent the first half of his childhood in Georgia, and not in Russia. They believe that later, as the head of the domestic intelligence agency, Putin changed his life story and denied the existence of his biological mother in order to speed his path to power – and to avoid being seen, during his first Russian election campaign, as an illegitimate child who had grown up in Georgia.