As the accusations and counter-accusations have flown around Washington this week, one man is, as ever, the centre of attention. The talk should be about his first overseas trip – a crucial one which will see the President visit the Vatican and the Middle East.

Instead, Donald J Trump's name is on everyone's lips thanks to the story that refuses to go away – the extent of his links with Russia.

During the campaign, Trump managed to dodge numerous issues that would have sunk other candidates. He even joked about this remarkable Teflon tendency, saying, 'I could stand in the middle of Fifth Avenue and shoot somebody and I wouldn't lose any voters.'

However, with the appointment of a special prosecutor to investigate the Russia links, whispers are beginning that Trump's seeming invincibility may be over.

The whispers don't only talk about the possibility of Trump's demise. They've begun to talk about the man who may replace him – Vice President Mike Pence. The rumours are getting so strong that the conservative publication National Review has published an open letter to Pence, calling on him to show loyalty to Trump.

Reports suggest Pence is beginning to attract considerable support within the beltway. His popularity among the Republican base was never in doubt, though. Among that base is a large contingent of evangelicals, of course.

More than eight in ten – 81 per cent – of white evangelicals voted for the Trump-Pence ticket in November. While much of the commentary around that focused on the nomination of Neil Gorsuch to the Supreme Court, it would be unwise to overlook the role that Pence's presence on the ticket played. Personally, having been brought up Catholic and become an evangelical, Pence has credentials with both groups – a notable advantage in Republican politics.

Pence was the insurance policy for evangelicals and conservative Catholics. Yes, Trump was a thrice-married casino owner, who'd boasted about being a sexual predator. But here was a good upstanding Christian man who would rein in Trump's worst excesses. It had been tried before of course. Running in 2008, Senator John McCain needed a boost to his evangelical credibility. His selection of Sarah Palin now looks, at best, eccentric. But the same logic drove the Trump team to select Pence.

Unlike Palin, the Governor of Indiana had impeccable credentials for those seeking a particular brand of small-state, low-tax Republicanism mixed with very conservative positions on social issues. His opposition to abortion and gay marriage meant that he passed the test required by many of the conservative evangelical voters who make up around a fifth of the entire US electorate. These shibboleths gave evangelicals some comfort – Trump may be a loose cannon, but Pence was a steady hand on whom they could rely.

Sure enough, in the days after the inauguration of Trump, Pence sent out a very clear signal to those who'd helped him become Vice President. He addressed a large anti-abortion rally in Washington, immediately making him the most senior government official to ever address the March for Life. The message was obvious – here is a leader upon whom conservative evangelicals can rely.

Of course, the other side of the aisle wasn't so sure. Pence, to some Democrats, was actually a more worrisome prospect that Trump himself. He had made enemies with his notorious religious freedom law in Indiana. His record as the most conservative candidate to run for VP in 40 years made him unpalatable to many, while his reluctance to accept Syrian refugees in Indiana was heavily criticised.

While Trump finds ever more ingenious and baffling ways to break conventions and to surprise Washington-watchers, Pence is the opposite. Opponents may be concerned about his agenda, but he is a career politician whose track record is well-known and who would most likely legislate accordingly. Would those opponents consider the old maxim that it's 'better the devil you know than one you don't?'

Trump remains in the White House, for now. But is there a path to power for the man who would be President? Maybe not... 'In the unlikely event that Trump is removed from office,' speculated Politico, 'Pence would assume the presidency amid a constitutional crisis. He could also be considered tainted by his past devotion to Trump.'

Yet, the whispers continue to grow with every passing hour and every passing tweet from the incumbent. Trump has survived so far but he may have met his match in the form of the man who was brought on board to make him electable in the first place.

Follow Andy Walton on Twitter @waltonandy