EU leaders have agreed to make “swift” progress on raising the tax bills for digital giants such as Google and Facebook, despite warnings from smaller states that unilateral action could drive business away from Europe.

France is pushing for a new way of taxing them on the basis of their turnover – rather than profits – to increase the amount collected from companies accused of paying too little in Europe.

Currently, such companies tend to be taxed on profits booked by subsidiaries in low-tax countries such as Ireland even when their revenues originate from other EU states.

It emerged from a meeting of finance ministers on Saturday that the French proposal has gathered the support of 10 of the 28 EU governments, including Germany, but the move is likely to need the backing of all member states to reduce risks of legal challenges.

Estonia, which holds the European council’s rolling presidency, has proposed pursuing a global solution with the cooperation of other economic blocs over a longer time period to avoid hampering the digital economy in the EU.

The chancellor Philip Hammond reportedly warned at the meeting against angering Washington, which could abandon global tax reform in retaliation. Further details of both proposals put forward in Tallinn will be fleshed out before a summit of EU leaders on 29 September, which is dedicated to digital issues, with a “direction” to be agreed at a summit of finance ministers in December.

However, it appears there is growing support for a “quick fix” within the bloc as a precursor to a global agreement. One EU finance minister at the meeting in Tallinn told his colleagues: “When traditional businesses are asked to pay their tax, they are not given extra time.”

The Dutch finance minister Jeroen Dijsselbloem called France’s plan “a very good initiative”. Belgium also supported the move, although its finance minister Johan Van Overtveldt said technical work was necessary.

Among the dissenting voices, was Denmark’s finance minister, Kristian Jensen, who said: “I think we should be very careful not to tax what we are going to live on in the future. I am … always sceptical by new taxes and I think Europe taxes heavily enough.”

Luxembourg’s finance minister, Pierre Gramegna, who acknowledged there was an issue with online giants’ taxation, said a tax on turnover would hit loss-making companies which are otherwise exempted from paying. He added that it “does not make any sense” for Europe to move without a global agreement. The Czech Republic and Malta both said technical work on a turnover tax would be very complicated.

France’s finance minister, Bruno Le Maire, urged the EU commission, which is in charge of making legislative proposals, to produce a formal text by mid 2018.

A statement from the Estonian presidency said ministers had agreed to move forward swiftly and to reach an understanding at the finance ministers’ summit in December. “For us, it is important to agree on new international tax rules that also take into account the business models of the digital economy,” said Toomas Tõniste, the Estonian minister for finance, after the meeting. “This would guarantee the equal taxation of all companies regardless of their location or place of activity. I hope today’s discussion helped us get a step closer to a suitable solution.”