(CNN) House Democrats are publicly holding out hope that historic impeachment hearings will persuade a vast majority of the American public that President Donald Trump committed high crimes and misdemeanors — but privately many acknowledge that it's unlikely to happen.

In a private meeting this week, House Speaker Nancy Pelosi and her top lieutenants were skeptical about the prospects of a dramatic shift in opinion even as public impeachment hearings began this week, according to multiple sources familiar with the matter. The upshot, the sources said: Democrats need to move forward with impeachment proceedings even if the politics are murky, noting that even during Richard Nixon's presidency most of the public was divided until soon before he was forced to resign.

"Well, I think there are hard views on both sides," House Majority Leader Steny Hoyer told CNN when asked if he thought the public's views would shift dramatically. "And sadly, apparently, Trump was perhaps right when he said of his own supporters that he could shoot somebody in the middle of Fifth Avenue and they would not require any accountability."

Democrats have made a concerted effort to shift their rhetoric to clearly convey to the public a more concise message: That Trump engaged in " bribery ," which is cited in the Constitution on impeachment, when he pushed Ukrainian officials to investigate his political rivals in exchange for $400 million in military aid the country desperately sought — a message Pelosi conveyed on Thursday, a day after the first public impeachment proceedings.

Yet Democrats say there's still a challenge in convincing a large majority of the public that what Trump did amounts to an abuse of power.

"Abuse of power is not necessarily a concept that most Americans run around thinking about," said Rep. Jim Himes, a Connecticut Democrat who sits on the House Intelligence Committee. "The point is we are all working to try to make a fairly unusual concept to most Americans — abuse of power — understandable."

Democrats kicked off their first impeachment hearing on Wednesday with two career diplomats, who gave a sobering account of how they learned that the US security aid and a one-on-one meeting was conditions on Ukraine announcing political investigations. Democrats described the hearing as a damning account, but Republicans were dug in that the testimony was hearsay and Trump did nothing wrong when the aid ultimately was provided to Ukraine.

Pelosi was hesitant to launch an impeachment inquiry over the allegations stemming from former special counsel Robert Mueller's investigation. But as the Ukraine scandal unfolded in September before she announced the impeachment inquiry, there was a significant shift in polling where a majority of the public supported impeachment.

Since then, however, polling has largely remained split, with a narrow majority backing impeachment. Between mid-October to mid-November, 47% of voters supported impeaching and removing Trump, compared to 44% who were against it, according to a CNN average of polls.

The discussion comes at a critical time for House Democrats as they prepare for the next phase of their proceedings: How to structure articles of impeachment against Trump.

Some Democrats want to include elements of obstruction of justice -- namely the 10 episodes of the President seeking to undercut the Russia investigation as detailed by the Mueller report -- while others want to keep the matter more narrowly focused on what they view as an abuse of power over Trump's handling of Ukraine.

Yet Democrats also worry not including articles of impeachment related to obstruction of justice will excuse the President's actions to allegedly thwart the Mueller probe.

"There was a lot of illegal behavior by the President outlined in the Mueller report," said California Rep. Ted Lieu, a member of the House Judiciary Committee. "But you have to balance that against not making it confusing for people."

On Thursday, Pelosi made clear that she didn't believe that concerns over obstruction of justice alone were enough to use as the basis of an impeachment inquiry.

"The clarity for the public to understand what is there wasn't as clear in my view when you say obstruction of justice, obstruction of justice, obstruction of justice, eleven times in the Mueller report," Pelosi said. "That isn't justification enough for inquiring into an impeachment."

Yet Pelosi also acknowledged that the public might not be there.

"This is very prayerful because impeaching is a divisive thing in our country. It's hard," Pelosi said. "And the place that our country is now, it's not a time where you go to 70% when people, when President Nixon walked out of the White House. It wasn't there before he left. Even two weeks before he left."

A Democratic aide said in today's political climate, there's "too much polarization" and Trump's base is unlikely to move, making it harder to shift public opinion.

At the meeting earlier this week, Pelosi made a similar comment to her top lieutenants, according to the sources, as the Democratic Campaign Committee Chairwoman Cheri Bustos of Illinois was discussing messaging and focus groups the campaign arm was conducing over impeachment. Pelosi has also made clear that the party needs to keep the focus on its legislative agenda, as they plan to push through the President's new trade deal with Mexico and Canada by year's end.

How impeachment plays in key districts has been an issue that Democrats have discussed repeatedly behind closed doors. At a meeting in late October, DCCC officials discussed focus group research that showed that voters wanted the inquiry to be a fact-finding mission — not a foregone conclusion, according to a source in the room.

And publicly, Democratic leaders have tried to make the case that they still have an open mind on impeachment, even though it appears all but certain that Trump will become the third President in history to be impeached.

House Judiciary Chairman Jerry Nadler, a Democrat from New York, said Wednesday's public hearing was "pretty damning." But when asked if he had enough evidence to impeach, Nadler said: "I'll keep my mind open at the moment."

Democratic lawmakers said the hearing was a success, even if it didn't create a dramatic moment, because it laid the groundwork for a compelling case that the President abused his office. And several Democrats said they aren't ready to abandon the notion that the public opinion could shift decisively to their side.

"It's divided, but we have to do our jobs. And I just always believe that the public will get there," said Rep. Val Demings, a Florida Democrat on the House Intelligence Committee.

For Republicans, the first impeachment hearing was also considered a success. Republican aides working on impeachment acknowledge that a primary goal over the course of the next several weeks is keeping their members together.

"Unity is key," one senior GOP aide working on impeachment said, citing the unanimous GOP opposition to the impeachment resolution adopted last month. "If we keep our guys together we send a clear message that this whole thing is partisan."