Update - 3.20pm: US intelligence officials have not publicly raised the possibility of contacts between the Clintons and Moscow.
Officials investigating the matter have said they believe Moscow had hacked into Democrats' computers in a bid to help Mr Trump's election bid.
National Security Agency director Michael Rogers later told the committee that the intelligence community stands behind its January assessment that it is highly confident Russia interfered in the 2016 election with the goal of electing Mr Trump.
Mr Rogers said that his agency is working to provide congress with the material it needs to investigate the intelligence agencies' findings.
Both Mr Comey and Mr Rogers said they have no evidence or intelligence that Russian cyber actors changed vote tallies in key states during last year's poll.
They both said they have no evidence that any vote tallies were changed in Michigan, Pennsylvania, Wisconsin, Florida, North Carolina or Ohio.
Mr Trump took to Twitter before the hearing began, accusing Democrats of making up allegations about his campaign associates' contact with Russia during the election.
He said Congress and the FBI should be going after media leaks, and maybe even Hillary Clinton, instead.
Mr Trump also suggested, without evidence, that Mrs Clinton's campaign was in contact with Russia and had possibly thwarted a federal investigation.
What about all of the contact with the Clinton campaign and the Russians? Also, is it true that the DNC would not let the FBI in to look?— Donald J. Trump (@realDonaldTrump) March 20, 2017
Update 2.59pm: Mr Comey told the US House Intelligence Committee: "This work is very complex, and there is no way for me to give you a timetable for when it will be done."
Earlier in the hearing, the chairman of the committee contradicted an earlier assertion from Mr Trump by saying that there had been no wiretapping of Trump Tower.
However, representative Devin Nunes, a California Republican whose committee is one of several investigating, said that other forms of surveillance of Mr Trump and his associates have not been ruled out.
National Security Agency director Michael Rogers will also address the hearing later.
Mr Trump tweeted: "The real story that Congress, the FBI and others should be looking into is the leaking of Classified information. Must find leaker now!"
The real story that Congress, the FBI and all others should be looking into is the leaking of Classified information. Must find leaker now!— Donald J. Trump (@realDonaldTrump) March 20, 2017
Update 2.47pm: FBI director James Comey has confirmed the agency is investigating possible links and co-ordination between US president Donald Trump's associates and Russia.
Update 2.41pm: There was no physical wiretapping carried out on Trump Tower, but it is possible that "other surveillance activities" were used against US president Donald Trump and his associates, the chairman of the US House intelligence committee said.
California representative Devin Nunes spoke at the opening of the committee's first public hearing on Russia's interference in the 2016 election.
The Republican said the committee has seen no evidence to date that officials from any campaign conspired with Russian agents, but will continue to carry out investigations into the issue.
House Intel Committee Chair Devin Nunes: No "wiretap of Trump Tower," but "still possible" other surveillance of Trump may have taken place pic.twitter.com/xq21G4msxX— PBS NewsHour (@NewsHour) March 20, 2017
He also said the committee will investigate who has been leaking classified information about investigations into Russia's interference.
Mr Nunes said he hopes the committee's hearings will result in a "definitive report" on Russia's involvement in the presidential election.
US president Donald Trump has accused Democrats of making up allegations that Russia interfered in last year's election, and said US Congress and the FBI should be focusing on media leaks instead.
His tweets came just hours before a potentially politically damaging hearing in which FBI director James Comey and National Security Agency director Michael Rogers plan to testify over allegations of Russian hacking and whether there were any connections between Moscow and Mr Trump's campaign.
Mr Trump tweeted today: "The Democrats made up and pushed the Russian story as an excuse for running a terrible campaign. Big advantage in Electoral College & lost!"
"The real story that Congress, the FBI and others should be looking into is the leaking of Classified information. Must find leaker now!"
James Clapper and others stated that there is no evidence Potus colluded with Russia. This story is FAKE NEWS and everyone knows it!— Donald J. Trump (@realDonaldTrump) March 20, 2017
A hearing before the House Intelligence Committee, one of several congressional panels probing allegations of Russian meddling, could allow for the greatest public accounting to date of investigations which have shadowed the Trump administration in its first two months.
The two most senior members of the House intelligence committee said yesterday that documents the Justice Department and FBI delivered late last week offered no evidence that the Obama administration had wiretapped Trump Tower, the president's New York City headquarters.
But the panel's ranking Democrat said the material offers circumstantial evidence that American citizens colluded with Russians in Moscow's efforts to interfere in the presidential election.
"There was circumstantial evidence of collusion; there is direct evidence, I think, of deception," California representative Adam Schiff said on NBC's Meet the Press.
''There's certainly enough for us to conduct an investigation."
Devin Nunes, the California Republican who chairs the committee, said: "For the first time the American people, and all the political parties now, are paying attention to the threat that Russia poses.
"We know that the Russians were trying to get involved in our campaign, like they have for many decades. They're also trying to get involved in campaigns around the globe and over in Europe."
The Senate Intelligence Committee has scheduled a similar hearing for later in the month.
It is not clear how much new information will emerge on Monday, and the hearing's open setting unquestionably puts Mr Comey in a difficult situation if he is asked to discuss an ongoing investigation tied to the campaign of the president.
At a hearing in January, Mr Comey refused to confirm or deny the existence of any investigation exploring possible connections between Mr Trump's associates and Russia, consistent with the FBI's long-standing policy of not publicly discussing its work.
His appearances on Capitol Hill since then have occurred in classified settings, often with small groups of representatives, and he has made no public statements connected to the Trump campaign or Russia.
But Mr Comey may feel compelled to respond to Mr Trump's unproven Twitter assertions that former president Barack Obama ordered a wiretapping of Trump Tower during the campaign.
Congressional leaders briefed on the matter have said they have seen no indication that that is true, and Mr Obama's top intelligence official, James Clapper, has publicly called the claims false.
The US justice department's disclosure on Friday that it had complied with congressional demands for information regarding Mr Trump's wiretapping tweets could allow Mr Comey to avoid questioning by simply saying that the representatives already have the information they requested.
Yet any lack of detail from Mr Comey will likely be contrasted with public comments he made last year when closing out an investigation into Hillary Clinton's email practices and then, shortly before election day, announcing that the probe would be revived following the discovery of additional emails.