The UK's secretary of state for culture, media and sport is to ask that search engines and other businesses must "make it more difficult" for piracy sites, the Financial Times has reported.
Jeremy Hunt is to outline his suggestion in a speech at the Royal Television Society's Cambridge Convention.
If no industry-led solution can be found, the government is prepared to legislate in the forthcoming Communications Bill.
If a court rules a site is essentially unlawful, the government would like to see search engines push it down their listings, while choking revenues from advertising or payments to make it unviable.
Television, music and movie companies have long pressed Google to prioritise legal services in its search results, but the proposals are likely to dismay open-internet advocates, who fear harm to free speech or competition.
Until now, British efforts to stem piracy have largely focused on internet service providers, while the Hargreaves review earlier this year recommended reforming intellectual property law to encourage more legal digital services.
Although the policy remains at an early stage, Mr Hunt is considering the creation of a new cross-industry body, which would identify piracy sites to be put through a streamlined legal process, allowing the courts to grant injunctions in a matter of days or weeks.
Over the last year, Google has stepped up its efforts to respond to legitimate requests from rights holders to remove links to infringing web pages.
It says on average it responds to “reliable” takedown requests in four hours, but it usually refuses to block entire sites, only specific pages.
Hunt is expected to say: “We intend to take measures to make it more and more difficult to access sites that deliberately facilitate infringement, misleading consumers and depriving creators of a fair reward for their creativity.”