The House select committee investigating the US Capitol attack on Monday asked three Republican members of Congress to assist its inquiry, as it seeks to establish the extent of their roles in Donald Trump’s attempt to overturn the 2020 election.

The panel sent letters requesting voluntary cooperation to Mo Brooks, Andy Biggs and Ronny Jackson, three congressmen who strategised ways to stop certification of Joe Biden’s election win or appeared to have connections to elements involved in the Capitol attack.

Bennie Thompson, the Democratic committee chair, told the Guardian last week the panel wanted to conduct interviews with Republicans so it could consider their testimony for its report, due to be published in September.

The panel opted against issuing subpoenas compelling testimony in the first instance, since that could cause the Republicans to attack, whereas an informal interview might at least yield some information, two sources close to the matter said.

In a statement, Jackson refused to cooperate.

“I will not participate in the illegitimate committee’s ruthless crusade against President Trump and his allies,” he said, hours after a Trump-appointed judge ruled in separate January 6 litigation that the panel was, in fact, legitimate.

House investigators are expected to issue further letters to Republican members of Congress, sources said. The Guardian first reported the committee was considering letters to Marjorie Taylor Greene, Lauren Boebert and Paul Gosar, in addition to Biggs and Brooks.

The panel made a particularly expansive request to Biggs, the former head of the ultra-conservative House Freedom Caucus, outlining four lines of inquiry that investigators want to pursue with respect to his role in the weeks before 6 January.

Thompson told Biggs the committee wanted to ask him about a crucial 21 December meeting at the White House between Trump and dozens of Republicans, which produced a plan to have the then vice-president, Mike Pence, refuse to certify Biden’s win.

The letter also said the panel wanted to ask what Biggs knew of plans to stage a march from the Ellipse, near the White House, to the Capitol on 6 January, through his purported contacts with the pro-Trump activist Ali Alexander, who led the “Stop the Steal” movement after the 2020 election.

Biggs’s potential contacts with Alexander are of special interest to the investigation, the sources said.

The committee is trying to untangle claims by Alexander that he “schemed up putting maximum pressure on Congress while they were voting” with Brooks, Biggs and Gosar, and his testimony that he spoke to Biggs’s staff and the congressman himself.

Alexander obtained a permit for a 6 January rally near the Capitol. But the rally never took place – although a separately organised event did – and Alexander was instead filmed going up the Capitol steps with members of the Oath Keepers militia group.

Thompson said the panel also wanted to ask Biggs about his efforts to pressure legislators to create “alternate” slates of electors for Trump in states he lost, as well as an alleged request he made to Trump for a pardon in the days after the Capitol attack.

In a letter to Brooks, the select committee said it wanted to ask him about his recent public comments, how Trump supposedly asked him on several occasions to “help rescind the election” and that “we’ve got to take Joe Biden down and put me in now”.

In the letter to Jackson, the panel indicated they would ask Trump’s former White House physician about why his name came up in encrypted text messages among members of the Oath Keepers as the Capitol attack was under way.

According to text messages in a recent court filing, an unidentified Oath Keeper messaged a group chat in the afternoon of 6 January that “Ronnie [sic] Jackson (TX) office inside Capitol – he needs OK help. Anyone inside?”

The same Oath Keeper provided an update less than 10 minutes later: “Dr Ronnie Jackson – on the move. Needs protection. If anyone inside cover him. He has critical data to protect.”

The Oath Keepers leader, Stewart Rhodes, responded: “Give him my cell.”

The extraordinary exchange raises the notion that members of the Oath Keepers – two of whom are among those who have been indicted on charges of seditious conspiracy for storming the Capitol – were close enough to Jackson to know his whereabouts and condition in real time.

To that extent, the letter to Jackson said, the select committee wants to ask him about whether he had contacts with the Oath Keepers, how the militia group could have known that he needed protection or, in another text, knew he had “critical data to protect”.