Andrea Leadsom reportedly made the comments to Saturday's Times newspaper after Theresa May spoke, in an interview earlier this week, of how she and her husband were unable to have children.
The two women are battling it out to replace David Cameron as Conservative prime minister and party leader he resigned following last month's vote to leave the European Union. The winner will be announced on September 9.
Leadsom, who is married with two sons and a daughter, was quoted as saying that May "possibly has nieces, nephews, lots of people.
"But I have children who are going to have children who will directly be a part of what happens next," she added.
She reportedly also said: "Genuinely, I feel that being a mum means you have a very real stake in the future of our country, a tangible stake."
After the comments were published, Leadsom tweeted a link to the story, saying: "Truly appalling and the exact opposite of what I said. I am disgusted."
Conservative party members will vote to decide whether May, who is ahead in polling, or Leadsom will become the second woman ever to lead Britain after Margaret Thatcher, who stepped down in 1990.
Conservative MP David Gauke, who is backing May, wrote on Twitter: "I'd like to think this is a case of verbal clumsiness not calculation. If the latter, yuk... Either way, an apology is due."
Journalist Piers Morgan tweeted simply "Wow" in response to the story.
And a former senior advisor to Cameron, Andrew Cooper, added: "Theresa May told Tories in 2002 they were 'nasty party'. David Cameron spent 11 yrs trying to fix it. Andrea Leadsom = nasty party again."
The comments were published as May issued a statement making a "clean campaign pledge" and urging Leadsom to do the same.
This included promising to ensure that campaigning remains "within the acceptable limits of political debate".
