The committee is also looking for ways to obtain documentation of Trump's conversations with top intelligence and law enforcement officials involved in a separate federal Russian Federation probe.
"We will be following up with subpoenas, and those subpoenas will be created to maximise our chance of getting the information that we need", Representative Adam Schiff said at an event sponsored by the Christian Science Monitor on Wednesday.
Former National Security Adviser Mike Flynn, already the subject a Senate Intelligence Committee subpoena, is now being subpoenaed by the House of Representatives as well.
Flynn's attorneys argued that the Senate's request was too broad, and if Flynn were to comply, he could be confirming the existence of some documents and, in effect, providing testimony that could be used against him.
The former national security adviser first came under scrutiny after it was revealed that he had spoken with the Russian ambassador, Sergey Kislyak, about United States sanctions against the country.
The Justice Department says Attorney General Jeff Sessions did not disclose on a security clearance form meetings he had as a senator a year ago with foreign dignitaries, including the Russian ambassador.
Both the Senate and House committees are hoping that Mr Flynn will provide information relating to the Trump campaign's possible ties to Russian Federation, and whether they engaged in any collusion during the 2016 presidential election.
Trump later fired Flynn for reportedly lying about his contacts with Russian Ambassador Sergey Kislyak.More news: Trump in Mideast condemns 'evil losers' behind Manchester attack
More news: Iran President Hassan Rouhani says missile tests will continue 'if necessary'
More news: JR Smith: LeBron James Must 'Play Confident' in Game 4
In addition to the congressional scrutiny, Flynn is now a target of an FBI counterintelligence investigation, a federal probe in Virginia and a Defense Department inspector general's inquiry into the propriety of foreign payments he accepted.
The intelligence panels are among several congressional committees looking into the allegations involving links between Russian Federation and Trump's November 8 victory.
The committee sent a letter to Flynn's lawyer Tuesday addressing concerns that its original subpoena lacked specificity.
Briefing reporters after a closed-door intelligence meeting on Tuesday, Sens. "The Fifth Amendment privilege does not apply to business entities, period", he said, adding that both Supreme Court and District of Columbia Circuit Court rulings would weigh on the committee's side. The U.S. Justice Department announced last week that former FBI Director Robert Mueller had been appointed special counsel to lead an independent probe of the matter.
Former Trump campaign foreign policy adviser Carter Page says he will testify next month before the House intelligence committee as part of its probe into Russia's meddling in the 2016 presidential election.
Flynn, a retired general, is a key witness in the Russian Federation investigations because of his ties to Moscow.
"That could be further evidence of an attempt to interfere with the investigation", Schiff said of Trump's reported request to Coats and Rogers.