WASHINGTON — President Donald Trump wants the House to vote on impeachment as soon as possible — ideally, three weeks ago.

Speaker Nancy Pelosi? Not so much.

The White House put proof of its stance in an eight-page letter on Tuesday evening — half lawyerly complaint, half campaign fundraising message — accusing the Democratic-led House of running a sham investigation and announcing Trump would block any further participation by his administration.

The letter challenged the rights of the House to set the rules of impeachment, charged Democrats with trying to reverse the results of the 2016 election and influence the 2020 contest, and concluded that there is "no legitimate basis" for the inquiry Pelosi is calling "impeachment" that's already underway.

"For the foregoing reasons, the President cannot allow your constitutionally illegitimate proceedings to distract him and those in the Executive Branch from their work on behalf of the American people," White House Counsel Pat Cipollone wrote.

The Constitution vests the House with "the sole Power of Impeachment" in Article I, Section 2, and prescribes in Article I, Section 5, that "each House may determine the Rules of its Proceedings." What's really going on is a fight over when the House will first take a vote involving Trump that has any form of the word "impeach" in it.

Though Cipollone's letter stops short of demanding it, Trump and his allies are trying to force Pelosi to do that sooner rather than later — with a full House vote to launch a formal "impeachment inquiry." The speaker, House Democratic sources have long said, would prefer to vote on actual articles of impeachment after an inquiry, which she created without a roll-call vote, wraps up.

There are some substantive and procedural reasons for Trump and Pelosi to take their respective positions, but this battle is really over gaining the upper hand in the politics of impeachment — both inside Washington and in shaping its effect on the 2020 elections.

For Trump, it's a race against the clock. For Pelosi, it's a matter of building momentum.

Public support for an impeachment investigation — and for removing Trump from office — has grown since it was revealed that he asked Ukraine to open an investigation into former Vice President Joe Biden, who is one of the Democrats running for the right to face him in 2020.

In an NBC/Wall Street Journal poll released Tuesday, 51 percent of Americans said allegations arising from the Ukraine scandal are serious and should be investigated, and a combined 55 percent said either there is enough evidence to remove him from office now or that an inquiry should be conducted to determine whether he should be impeached and removed.

A Quinnipiac survey, also released Tuesday, found 45 percent support for removing Trump from office and 49 percent opposition among registered voters — a change from 37 percent in favor and 57 percent opposed on Sept. 25 and a minor deviation from a 47-to-47 split last week.

Full coverage: Trump impeachment inquiry

On the surface, Trump and Pelosi have engaged in an unlikely role reversal — a commander in chief demanding a move toward Congress' form of indictment and the leader of the opposition party holding back. But it's one that, after taking shape slowly for several months, has been super-charged by the matter of Trump's dealings with Ukraine.

The subtext of the fight has everything to do with the politics inherent to the impeachment process.

There are two factors, in a way at odds with each other, that are nonetheless entrenching their respective positions. On the one hand, it's Pelosi who has more to lose with any politically tough vote because her party holds the majority, and Republicans want her incumbents who represent districts Trump won to go on the record now and as often as possible.

"It's not popular" in Democratic-held districts that Trump won, said Michael Steel, who served as an aide to former House Speaker John Boehner, R-Ohio. "They think the Democrats in swing districts will suffer blowback."

And Trump would like to see a party-line vote, which could bolster his argument that Democrats are dividing the country unnecessarily for political gain.

On the other hand, public opinion is shifting away from the president, meaning it's getting easier for rank-and-file Democrats to vote in favor of impeachment and perhaps a bit harder for a handful of Republicans to vote against it. The sooner Trump gets a vote, the less likely he is to see Republican defections. And, if he can force Pelosi to hold an impeachment inquiry vote, it would be extremely difficult for any Republican who votes against that to cast a vote for impeachment later.

Former Rep. David Jolly, who represented a Florida district as a Republican, said that along with the friction created by Trump and Pelosi's conflicting goals with regard to getting lawmakers of both parties on record, they are clashing over litigation strategy.

"The absence of a floor resolution allows the administration grounds to litigate and delay everything, as we're currently seeing," said Jolly, who is a lawyer. "But so, too, does a resolution that inevitably would limit minority procedural rights around subpoenas and disclosure of material evidence, creating a second round of litigation and delay."

While Pelosi doesn't want to lose momentum toward actually impeaching the president, she is also in no rush to force a vote on launching an inquiry when she already has one that has produced big headlines. Some Democrats are convinced that polling numbers will move in their direction. Even if there's no clear-cut call for the president's ouster right now, they believe new evidence could sway voters.

On Tuesday, Pelosi told reporters that the probe should be conducted "carefully, not hastily" — which is a far cry from how she described the process to House Democrats on Sept. 24, when she told them at a closed door meeting that she wanted to move "expeditiously."

The two sides are involved in a court battle over whether the administration can block the House from obtaining grand jury testimony that was part of former special counsel Robert Mueller's Russia probe because, as the Justice Department argues, the House has not voted to start a formal impeachment inquiry. The House's position is that it has the sole power to determine when it is conducting an impeachment investigation.

Trump could have jump-started the process — formal or informal — at any point if he had allowed administration officials to testify and turn over documents that House Democrats requested and, in some cases, subpoenaed.

And Cipollone left the White House an exit ramp if Pelosi were to accede to the president's demands.

"If the committees wish to return to the regular order of oversight requests, we stand ready to engage in that process," he wrote.

But it was hard to see those words, cloaked as they were, in a chest-thumping letter that couldn't have been more clearly designed to rile up partisans on both sides and pressure Pelosi to act on impeachment quickly.

She was unmoved.

"The White House should be warned that continued efforts to hide the truth of the president’s abuse of power from the American people will be regarded as further evidence of obstruction," Pelosi wrote in a response Tuesday. "Mr President, you are not above the law. You will be held accountable."