United States officials and members of Congress have also taken hard looks at some big American companies’ tax arrangements in Europe. And the Treasury Department has tried to block so-called tax inversions, in which American companies use mergers to relocate to European countries as a way to reduce their tax bills — although the rules might not be able to stop the drug giant Pfizer from using its acquisition of much-smaller Allergan to relocate its headquarters to Ireland.

Although the British prime minister, Mr. Cameron, defended his Google deal, his political opponents have criticized it as too generous to the company and not reflective of the full magnitude of its British revenue for the 10 years it covers.

In studying the Cameron-Google agreement, Ms. Vestager may follow up on a request of the Scottish National Party, a British opposition party, which has sent a letter to the European Commission criticizing the tax deal with the search giant. A spokesman for Ms. Vestager’s office said on Thursday that the complaint would need to be studied to see if it warranted a formal investigation.

The issue could be particularly delicate, coming as Mr. Cameron has been trying to negotiate better terms for Britain within the European Union. As soon as June, Britain may hold a referendum in which voters would be asked whether they want to remain in the bloc or leave it.

But Christian Cubitt, a Cameron spokesman, declined on Thursday to elevate the tax dispute to the level of the debate on Britain’s place in the European Union. He said simply that it was “a decision for the commission" to decide whether to investigate, although he stressed that Google had agreed to pay all the tax that is due.

Britain is by no means alone in seeking to handle corporate tax matters on its own terms.

Italian financial police officials were to issue an investigative report to Google on Thursday, indicating whether the company has paid sufficient taxes in Italy since 2008. Although Google has not been accused of wrongdoing, tax officials say the delinquent amount could be about 300 million euros, or about $325.3 million.