British homes and businesses will have a legal right to high-speed broadband by 2020, the government has announced, dismissing calls from the network provider BT that it should be a voluntary rather than legal obligation on providers.

The Department for Digital, Culture, Media and Sport said only a universal service obligation (USO) would offer certainty that broadband speeds of at least 10Mbps would reach the whole of the UK by 2020.

Broadband providers will now have a legal requirement to provide high-speed broadband to anyone who requests it, no matter where they are in the country.

BT had said it would pledge to voluntarily close the digital gap in broadband speeds between cities and rural areas, and would start work immediately, but it argued that secondary legislation could slow progress down.

However, the department said it did “not feel the proposal was strong enough for us to take the regulatory USO off the table, and have therefore decided not to pursue BT’s proposal, in favour of providing a legal right to broadband”.

The culture secretary, Karen Bradley, said she was grateful to BT for its proposal but had decided that only a regulatory approach would ensure high-speed broadband for everyone in the UK, regardless of where they lived or worked.

“We know how important broadband is to homes and businesses and we want everyone to benefit from a fast and reliable connection,” she said.

BT, which provides broadband to its own customers and to other suppliers via its Openreach network, said it respected the government’s decision.

“BT and Openreach want to get on with the job of making decent broadband available to everyone in the UK, so we’ll continue to explore the commercial options for bringing faster speeds to those parts of the country which are hardest to reach,” it said.

“We look forward to receiving more details from the government outlining its approach to defining the regulatory USO, including the proposed funding mechanism.”

The digital minister, Matt Hancock, said the rollout would not mean high-speed broadband was automatically delivered to every property. “It’s about having the right to demand it. It’s an on-demand programme. If you don’t go on the internet and aren’t interested then you won’t phone up and demand this,” he told BBC Radio 4’s Today programme. “The ‘access’ is being able to demand it.”

Hancock admitted the UK still lagged behind many parts of the world in terms of broadband speeds – in Japan, for example, 97% of connections have full-fibre lines, compared with 3% in the UK.

“This is the next big drive we have got to do as a country,” he said. “Our rollout of super-fast has been the fastest among comparable countries. The drive to get the full fibre connections, the future-proof connections, started only a year ago. I’m absolutely determined to see that rolled out.”

Earlier this month, Ofcom said more than 1m “forgotten homes” across the UK were unable to get sufficiently fast broadband to meet a typical family’s needs, such as streaming films or music.

The communications regulator said 4% of UK homes and offices, about 1.1m properties, could not access broadband speeds of at least 10Mbps.

Rural families were more likely to be left behind, with 17% of homes not receiving decent internet, compared with 2% in cities and towns.