The U.K.'s Information Commissioner's Office issued the order this week, giving Google 35 days to remove the links. Google has the right to appeal the order to the General Regulatory Chamber.
The order puts a "meta" spin on the controversial right to be forgotten ruling, which lets people request that Google remove links to information about them from its search results on its European sites.
The ruling, issued last year, established a mechanism for people to ask search engines to remove links to information they consider to be irrelevant or not in the public interest, though without removing the actual content from the Web.
Google's response to this week's order could show how wide a net the requests might be allowed to encompass.
The order was issued in response to a complaint from someone who had already had links removed around a conviction for a "relatively minor" offense from nearly a decade ago, the order said.
Google removed links to sites containing content related to the offense.
But, when links to news articles about the removal started to appear after a search for the person's name, the person asked Google to remove those links as well.
Google refused, the order said, on the grounds that the new links were relevant and in the public interest.
Google took into account the media's journalistic judgment in determining whether the information about the removals was relevant and in the public interest, the order said.
On Thursday, a Google spokeswoman said that the company had received the order and was reviewing it, but declined to comment further.
In the U.K.'s order, Deputy Information Commission David Smith acknowledged that the new links related to journalistic content.
"The commission does not dispute that journalistic content relating to decisions to delist search results may be newsworthy and in the public interest," he wrote.
"However, that interest can be adequately and properly met without a search made on the basis of the complainant's name providing links to articles which reveal information about the complainant's spent conviction," he wrote.
The order, in other words, does not want all the new links removed; only the links that appear after searches for the person's name.
Data leaked earlier this year by The Guardian showed that many of the removal requests submitted to Google so far have come from everyday people requesting the removal of links related to private or personal information.