In May last year, the European Court of Justice ruled that European residents can ask search engines to delete results that turn up under a search for their name when they are out of date, irrelevant or inflammatory - the so-called right to be forgotten.
Since then, Google and other search engines such as Microsoft`s Bing and Yahoo have begun to grant de-listing requests when they meet certain criteria.
But there has been much debate over the implementation, especially of Google`s decision only to scrub results from European sites, leading some to appeal to local regulators.
The company maintains it should only apply the ruling across its European domains, such as Google.fr in France and Google.de in Germany.
But EU data protection watchdogs, many legal experts and former German Justice Minister Sabine Leutheusser-Schnarrenberger, who has advised Google on privacy following the European ruling, think it should be global.
Some individuals have taken Google to court to try to force a change. They include Dan Shefet, a French lawyer born in Denmark, who won a defamation case in a French court recently that experts say called for the results to be scrubbed globally.
"In accordance with the (European court) judgment, the CNIL considers that in order to be effective, de-listing must be carried out on all extensions of the search engine and that the service provided by Google search constitutes a single processing," the CNIL regulator said.
France is the first country to open a potential sanctions process against Google if it does not change its position. But the powers of the CNIL remain limited, since it can only impose fines of up to 150,000 euros ($168,000).
The Mountain View, California-based Google had revenue of $66 billion last year.
A Google spokesman said the company had been cooperating closely with data protection authorities and was seeking the right balance in applying the European Court`s decision.
"The ruling focused on services directed to European users, and that`s the approach we are taking in complying with it," said the spokesman. ($1 = 0.8937 euros) (Reporting by Leila Abboud and Julia Fioretti; Editing by David Clarke)
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