The figure for 2015 is included in the breakdown of the UK's current account, known as the Pink Book, published by the Office for National Statistics (ONS).
The Leave campaign attracted criticism during the EU referendum for printing a slogan on the side of its battlebus that said: "We send the EU £350 million a week."
Supporters of the Remain campaign said the figure was misleading as it did not take into account the money the UK receives from the EU in the form of a rebate, as well as payments made to the public sector.
Boris Johnson stood by the £350 million figure on Thursday when interviewed on ITV's Good Morning Britain, saying it was "not disputed".
The ONS figures for 2015 suggest the UK's gross contribution to the EU, before the application of the rebate, totalled £19.6 billion - or about £376 million a week.
But, as the UK Statistics Authority pointed out during the referendum campaign, the Treasury pays the UK's contributions to the EU after deducting the value of the rebate.
The rebate in 2015 was £4.9 billion. Subtracting this from the gross contribution gives a figure of £14.7 billion.
A further subtraction of the EU's payments to the UK public sector gives the final figure of £10.4 billion, or about £199 million a week.
The precise amount of money the UK sends to the EU is difficult to calculate.
According to a study published by the ONS in May 2016: "None of the figures on the money flowing between the UK and EU are fully reflective of the total costs and benefits of EU membership as these are complex and difficult to quantify."
For example, European Union payments made directly to the private sector - such as universities and research organisations - are not included in these figures, as the ONS looks only at transactions directly between the EU and UK public bodies.
The breakdown for 2016 is due to be published later this year.