Apple last week reported that its cash mountain had grown to $178bn (£118bn) – among the biggest of any public corporation in the world. The lion’s share is surplus profits from outside America, in large part thanks to the huge popularity of its phones, laptops and other products around the globe, but also in no small measure because of its aggressive Irish tax structure.

As at the end of September last year, cash held offshore and beyond the grasp of the US taxman stood at $137bn and is likely to have grown since. At present, this sum cannot be reinvested in Apple’s US businesses nor returned to shareholders without incurring a colossal tax bill. In effect, it is in limbo.

The cash pile is technically owned by Apple’s subsidiaries in Knocknaheeny, a rundown northern suburb of Cork, Ireland. But it is managed by a rarely talked-of investment subsidiary closer to home and largely invested in American assets, including billions in corporate bonds and US government debt.

There is only one brief mention of this opaque internal investment company in Apple’s annual report, on page 114. It is called Braeburn Capital and registered in a quiet corner of Reno, Nevada. Some commentators have described it as “the biggest hedge fund you’ve never heard of”.

It is this kind of operation – common to many US multinationals, particularly in the tech and pharmaceuticals industries – that Obama is looking to deflate, freeing up America’s largest companies to repatriate cash held in limbo, but without releasing them entirely from US tax liabilities they have been avoiding for years, often very creatively.

One consolation for Apple investors is that they may be partially cushioned from the full impact of Obama’s proposals thanks to some cautious accounting provisions in recent years. The group has steadily built up prudent reserves against potential US tax bills on its accumulating offshore cash. Though this doesn’t reflect the US tax actually paid by Apple, these accounting reserves mean the group’s latest accounts show a lesser sum of $69.7bn ($54.4bn for 2013) as the surplus foreign profits Apple retains overseas, “indefinitely reinvested” beyond the reach of the US taxman.

Apple’s shares hardly reacted to Obama’s tax proposals in early trading in New York on Monday. The president’s comments outlined a resolution to the offshore cash problem that are not so far from measures advocated by Apple chief executive Tim Cook.

But while the move may prove an elegant solution to the central controversy in the US over how its multinationals are taxed, it is unlikely to go down well in Europe. Political leaders in France, the UK and Germany have grown increasingly vocal in their criticisms of US multinationals in recent years — angry in particular at how they have gamed different tax jurisdictions to achieve ultra-low tax rates on profits generated in Europe.

In the case of Apple, officials from the European commission have already signalled that they believe it has received such a generous tax deal from Ireland – where about two-thirds of Apple’s global profits accumulate – that it is in effect receiving illegal state aid. Apple’s Irish operations have been taxed at rates as low as 2%.

If the commission’s provisional view hardens into a formal conclusion later this year, those Apple subsidiaries in Cork that Obama has in his sights could be required to pay billions in European back taxes.

While the sums involved are enormous, there seems every prospect that all parties – investors, business leaders and tax officials around the world – will ultimately feel they have failed to get their fair share.