(CNN) The House Oversight Committee has instructed its Democratic members not to deviate from agreed upon lines of questioning at Michael Cohen's public hearing Wednesday, prompting grumbling from several Democrats on the panel who want to press President Donald Trump's former personal attorney about a range of matters related to Russia.

Those areas, outlined in a nine-page memo to the Democratic staff, include a range of Trump's conflicts of interests, Trump Organization business practices as well as payments to Trump's alleged mistresses to silence their stories before the 2016 elections.

But they do not include a range of topics about Russia, instead leaving those matters to Thursday's classified hearing before the House Intelligence Committee, an early indication of the tension bound to grow between several powerful House committees all investigating aspects of Trump's life.

Indeed, several Democrats on the panel told CNN that the public will get an incomplete picture if Russia topics are closed off, given that Wednesday's session will be the only public testimony Cohen will give out his of three congressional appearances this week.

"We want to hear about the relationship between Mr. Cohen and the President and candidate Trump, and so if he were inclined to have a lengthy discussion and an instructive discussion on, that then I'd like to have him proceed," said Rep. Stephen Lynch, a Massachusetts Democrat on the Oversight panel. "The way we have it sort of walled off here, I'm not sure that may be possible."

Asked about splitting the lines of inquiries between multiple committees, Lynch added: "I'm not sure that's the most effective way to do it."

The matter has been a topic of closed-door debate among committee Democrats.

According to several sources familiar with a private meeting Tuesday night of committee members and staff, lawmakers engaged in a back-and-forth over whether they can pursue lines of inquiries about Trump's ties to Russia, with some noting that there are a range of Russia subjects that could be broached that would not interfere with special counsel Robert Mueller's probe.

But those topics, including Cohen's admitted lies about the Trump Organization's pursuit of a Trump Tower Moscow project in the heat of the 2016 campaign, which were discussed at Tuesday's classified Senate Intelligence Committee hearing, will instead be relegated to a private hearing on Thursday before the House Intelligence Committee.

Virginia Rep. Gerry Connolly, a senior member of the Oversight panel, told CNN he asked Cummings and his staff at the Monday meeting about "the rules of the road." He said he's "pressing the envelope" to see what lines of inquiry could be asked Wednesday on issues related to Russia.

"Does that mean we can't ask about Russian mob money? Russian oligarchs? The Trump Tower in Moscow that he lied about? The Russian oligarchy purchase to Trump's Florida property at a highly inflated price? The fact that Russians own apartments in Trump Tower?" Connolly asked. "I mean, when you say Russia, is all of that off limits? And that was really the nature of it."

Connolly, who noted he's also a member of the House Foreign Affairs Committee, noted that for the past two years under GOP control they've little opportunity to question witnesses about Russia -- "and now here's the opportunity to maybe pursue some of that."

Last week, Oversight Chairman Elijah Cummings issued a public memo outlining the scope of the public hearing, including Trump's finances, his conflicts of interest, his foundation and his hotel as allowable topics. But the memo did not list topics pertaining to Russia, including the Trump Tower Moscow project that Cohen negotiated.

An expanded version of that memo was sent this week to Democratic members, along with the instructions to stay within the scope of the hearing.

In his memo, Cummings said that he consulted with Mueller, the Justice Department and House Intelligence Committee Chairman Adam Schiff, a California Democrat, in deciding the parameters for the public hearing.

"The scope of the Oversight Committee's open public hearing will not include questions relating to the Intelligence Committee's investigation of efforts by Russia and other foreign entities to influence the U.S. political process during and since the 2016 U.S. election," the memo says.

The memo adds that the public hearing also won't touch "the counterintelligence threat arising from any links or coordination between U.S. persons and the Russian government -- including any financial or other compromise or leverage foreign actors may possess over Donald Trump, his family, his business interests, or his associates." It goes onto say Cohen's lies to Congress about the Trump Tower Moscow project are not under the hearing's purview.

Asked Tuesday about some Democratic members eager to ask Cohen about Russia, Cummings declined to discuss the matter or the scope of the hearing.

"I'm not going to comment anymore," Cummings said.

On Tuesday, Schiff praised Cummings handling of the public hearing and defended the decision to split the topics between the committees.

"I think that's a good division of labor which will allow the public to get information about the President's conduct, but will also ensure that we can get complete answers to our questions in the intelligence committee and that we also respect the equities of the Justice Department and our own investigative needs," Schiff told CNN.

One Democratic source says that the agreement struck by Schiff and Cummings shows the "adults in the room" were able to successfully resolve legitimate differences over the overlapping interests of the committees.

Asked about Schiff's role, Connolly said: "It's my understanding that he has staked out a role for the intelligence committee that largely subsumes questions about Russia."