A committee of peers heard that as much as 85 per cent of the British market relies on Google's technology. Between them, Google and Facebook take around 55 per cent of every pound spent on digital advertising.
A CMA investigation would be likely to become a focus for calls to break up Google and Facebook by separating their monopolies on major web search and social networking from their sales operations.
Their opponents argue the move would encourage innovation and competition, and help publishers demand a bigger slice of digital advertising.
A senior advertising source suggested multiple sales houses could "plug into" the Google search algorithm, for instance.
Less radical action could curb the tech giants by ensuring they make it easier for advertisers to judge the impact of their spending.
Cairncross is considering such issues as part of a broad analysis of press funding and the impact of digital "clickbait" for Matt Hancock, the Culture Secretary.
As well as a competition investigation, it is understood she is looking at separate options for forcing Google and Facebook to share more of their advertising income with publishers.
Dozens of local newspapers have closed in the last 18 months and many surviving titles are operating a skeleton staff.
Rupert Murdoch has urged the introduction of "carriage fees", whereby news publishers would negotiate blanket fees with Google and Facebook as broadcasters do with pay-TV operators.
Government manoeuvres against Google and Facebook have been boosted by backing for CMA intervention via the Cairncross Review from Tom Watson, Labour's deputy leader.
A frequent critic of the press, he told the Daily Telegraph: "There's no doubt that the dominance of the big social media and internet companies has had a big effect on journalism and it has often been very negative.
"We need to find a way to safeguard traditional public interest journalism, and particularly local journalism, from being crushed beneath these expanding social media and internet giants. We need to scrutinise how the digital advertising market is working, or maybe not working. A CMA investigation would be the obvious place to do that."
He said Labour was considering whether to back a levy on Google and Facebook to help fund journalism.
Cairncross is understood to be grappling with how the press could be shielded from government influence if a levy was imposed. Labour Party leader Jeremy Corbyn has warned publishers that "change is coming" under a Labour government.
International calls have been mounting for action against the market power of Google and Facebook, intensified by political controversies over "fake news", hosting of terrorist and paedophile material and privacy violations.
Over the weekend, a joint statement from British, US and European publishers warned that "Google holds a vast share of the digital advertising market and there are very few alternatives", as part of a row over advertising contracts under Brussels' new GDPR regulations.
A spokesman for the publishing companies said: "Google is effectively putting a gun against publishers' heads. The implication in not signing Google's GDPR terms, and not doing business with them, is that publishers will suffer dire consequences in terms of revenues. This is a flagrant abuse of their dominant position."
Other players in the digital advertising sector are arguing that the stranglehold of Google and Facebook could be broken by Amazon, which has a fast-growing advertising business.
Steve Allan, the global chief executive of the WPP agency Mediacom, said that despite concerns, a full regulatory clampdown would be "premature".
Google and Facebook have recently redoubled attempts to work with publishers as regulation looms. In March, Google said it would spend US$300 million ($433m) to tackle misinformation online and help publishers build subscriptions.
Facebook has sought to back away from news. Its decision to show fewer news articles in its main feed in favour of personal updates from friends has hit outlets such as MailOnline, which last week revealed a 9 per cent drop in traffic.
