An entire floor of an obstetrics research centre was supposedly closed off and an Italian doctor flown in.
Newspaper Moskovsky Komsomolets published the story on its website, but it was afterwards taken down.
Another outlet, dni.ru, ran the headline: “Alina Kabaeva gave birth to twins and disappeared.”
Putin, 68, has repeatedly denied being in a relationship with the former gymnast.
Alina – dubbed “Russia’s First Mistress” – has also rejected the rumours.
Concern is growing that Alina is still missing, The Sun reports.
A Moscow source told the newspaper: “Putin is an intensely private man – he has been hiding his two adult daughters under fake IDs for years.
“Even now, though he talks about them occasionally, he never names them.
“If Alina gave birth to his children, her hiding away might be one of the strongest hints pointing to this..
“He is obsessive about the security of his family.”
Alina has won two Olympic medals, 14 World Championship medals, and 25 European Championship medals.
Following her glittering athletic career, she became an MP in Putin’s United Russia party.
Alina once also posed semi-naked in a magazine and attempted to break into singing.
She was most recently boss at National Media Group, which got a deal to broadcast Netflix content.
Her rumoured relationship with Putin was first reported on by the Russian newspaper Moskovsky Korrespondent in 2008, run by media tycoon and former KGB spy Alexander Lebedev.
The story was denied and the paper was shut down.
Rumours of another child surfaced in 2015, as the Russian leader went missing for more than a week.Alina was engaged to politician David Museliani in 2004 until their split a year later.
However, the couple divorced in 2013 and Putin has kept his private life guarded ever since.
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