Visitors to officialmikepence.com discover an elaborate spoof website for “Mike Pence: the 46th president of the United States”. A banner at the top asks: “Are you a homosexual? If yes, click here.” (It links to hell.com.) A campaign logo refers to Pence and his running mate Christ with crosses and stars. “President Mike Pence is proud to serve the white and Christian population of America for more than two decades,” the page says.

The notion of President Pence suddenly seems a lot less unlikely than it did a week ago. A special counsel has been appointed to investigate the Donald Trump campaign’s alleged collusion with Russia. The word “impeachment” was spoken on the floor of the House of Representatives and the “i-word” became an instant television cliche. Senator John McCain, musing that “we’ve seen this movie before”, invoked the Watergate scandal that led to the resignation of Richard Nixon and the succession of Vice-President Gerald Ford – who, like Pence, was a churchgoing conservative who earned his spurs in the House.

For Republicans, Pence could offer a much-needed antidote to Trump’s unpredictability: the steady hand, authentic conservatism and respect for Congress that they crave. For Democrats, the notion of bringing in a man who in January told an anti-abortion march, “Life is winning again in America”, may pose something of a dilemma.

“This is under the category of ‘careful what you wish for’,” said Bill Whalen, a research fellow at the Hoover Institution thinktank in Palo Alto, California. “What if you succeed in getting rid of Donald Trump and now you have a president who’s not up at four o’clock in the morning tweeting, a president who is not temperamental, a president who knows how to govern – he’s been the governor of a state – a president who knows how Washington works, a president who is self-disciplined?

“You might now have created your worst problem because that might end up being a very productive president. You also are putting somebody in who is very socially conservative. So whereas Donald Trump talks about social conservatism but basically he can be talked out of almost anything by his daughter, Mike Pence actually walks the walk.”

Whalen, a former chief speechwriter for the California ex-governor Pete Wilson, a Republican, added: “The Democratic goal here is not to impeach Donald Trump, at least not for the next two years. It’s just to critically wound Donald Trump, just try to create a Dunkirk for Donald Trump and keep him pinned down through the next midterm and get back the House.”

Donald Trump greets the then Indiana governor, Mike Pence, in July 2016. Photograph: Aaron P. Bernstein/Getty Images

From the election campaign to the White House, Pence, 57, has tried to present a public face of unwavering support for Trump while keeping his hands clean of the president’s most outrageous blunders. It has not always been easy. The former Indiana governor admitted that he could not defend Trump’s boasts about groping women when they came to light in an Access Hollywood video. His explanation for the dismissal of the FBI director, James Comey, was undermined by Trump himself.

Jim Brainard, a six-term Republican mayor from Carmel in Indiana, said “I can see the pain on his face” each time the vice-president has to deal with Trump’s antics. “I know the man and he’s not the sort of person who would deal recklessly with foreign policy or make statements on social media without thought or do anything that would jeopardise the country’s best interests. He thinks before he talks.”

In a spell of extraordinary turmoil even by Trumpian standards – his dismissal of Comey, revelation of sensitive information to the Russians and allegations that he tried to quash an FBI investigation into a senior aide – Pence has managed to remain largely under the radar. But not entirely. He was, after all, head of the Trump transition team in January when Mike Flynn informed transition officials that he himself was under FBI investigation for his ties to Turkey; Pence claims that he did not know that Flynn was a foreign agent until the news broke in March.

On Thursday night, the broadcaster Rachel Maddow of the MSNBC network devoted a segment to hammering him. “Vice-President Mike Pence has made a number of blunt, direct false statements related to the Trump-Russia investigation, both during the transition and since he has been vice-president,” she told viewers. “Mike Pence has his own troubles when it comes to this scandal.”

Ben Wikler, Washington director of the progressive group MoveOn.org, said: “Mike Pence has been on the Trump train. We don’t know if Trump colluded with Russia but we know that Pence colluded with Trump. The idea that he would emerge unscathed from the wreckage is not something that makes me lose any sleep.”

Even so, simple pragmatism on the part of Republicans could make Pence a tempting plan B if there are grounds for impeachment and the party suffers a meltdown in the 2018 midterm elections. A conventional, low drama, even-keeled figure in the White House would be bliss after the chaos of Trump that threatens to derail their legislative agenda.

George Ajjan, a Republican consultant and analyst, said: “Pence has outstanding relationships with the GOP leadership on Capitol Hill of all stripes, not just the social conservatives. So there’d be clear alignment and rapid progress on healthcare, taxation, and many other key policy initiatives that have eluded the party over the past months as a result of Trump’s unorthodox approach.

“The million-dollar question, of course, is how embattled Trump has to become before the ‘do you really want it’ idea is discreetly floated to Pence by his former House colleagues.”

Acquaintances say Pence wants the job, sooner or later, which partly explains why he served as governor of Indiana. On Wednesday, he quietly filed paperwork with the Federal Election Commission to set up a political action committee that will allow him to raise money for his political interests and make donations to down-ballot candidates.

Rich Galen, former press secretary to Dan Quayle, who was also from Indiana and served as vice-president under George HW Bush, said: “If I was advising Pence, I would be working very closely with the Republican House and Senate committees to set up my schedule to do fundraisers for people who might be in trouble because of Trump.

Melania Trump and Mike Pence embrace as they walk across the South Lawn of the White House on Friday. Photograph: Andrew Harnik/AP

“He has a good reputation among Republicans and he’s a pretty rightwing guy. If people think someone like Nelson Rockefeller [Ford’s vice-president, seen as moderate] is going to step in, they’ll be sadly mistaken. He’s got a demeanour that more people would be comfortable with.”

The 22nd amendment to the constitution, Galen noted, stipulates that if the vice-president takes over with more than two years left in the term, he can only run for re-election once. If he takes over with less than two years remaining, he can run twice, raising the prospect of Pence serving as president for nearly 10 years.

But a President Pence would struggle to get everything his own way. David Brady, a senior fellow at the Hoover Institution, cautioned: “It assumes a set of conditions that I view as about zero possibility. Trump’s impeached and the Republicans manage to hold the House and the Senate? That’s not going to happen. The voters would punish them and the result is you’ve got Pence as president and he’s more like Gerry Ford.”