The newspaper reported "a close friend" of the woman - with whom he has been pictured on social media - as saying "clearly there is a cultural problem within the team if this type of thing is going on".
She said the woman was devastated to find out Kaino was married, the newspaper claimed.
Jerome and Diana Kaino have three children.
In a statement, NZR said: "Information published concerning Aaron Smith in New Zealand media today is also a new development. NZR cannot comment at this stage on what are both personal matters, but we feel for all of those affected," NZR chief executive Steve Tew said.
"We certainly understand that our game and our players are under public scrutiny and these latest stories are concerning.
"As an employer, we will take some time to assess this. While NZR does not wish to comment on individual behaviour, it is clear that this is really hurting rugby, all the people concerned, our fans and supporters.
"We take these issues very seriously, and are also mindful that these issues affect people with families and loved ones.
"At a time when we are grappling with our cultural reputation, the Respect and Responsibility Review, which is currently under way, is even more important. The Review is being presented to the NZR Board this month and we are looking to make that public sometime in September."
The All Blacks play the Wallabies in a Bledisloe Cup match in Sydney tonight.
The Weekend Herald reported this morning that the huge distraction of Aaron Smith's toilet tryst at Christchurch Airport and its aftermath was continuing, with fresh revelations suggesting All Black management knew the star halfback's behaviour was not a one-off.
The Herald learnt of fresh allegations about Smith's actions - including claims he had sex with another woman several weeks after he left the All Black team in South Africa in disgrace last October following the toilet incident.
The woman's partner claims a member of All Black management rang him a few months later asking if they could help.
NZ Rugby announced this week that it is reopening an investigation into the original sex scandal, after a claim by the woman involved that Smith offered her money to sign an affidavit which would have falsely claimed they did not have sex in the airport toilet.
The Weekend Herald put questions about the latest allegations to NZR but they were not answered by its Friday night deadline.
Two men approached the Weekend Herald independently to tell their stories about Smith's alleged affairs with their partners.
Smith will still play in tonight's Bledisloe Cup opener against Australia in Sydney.
