It was a team born out of a political problem: Mr. Trump’s surprise march to the nomination had left the party’s establishment openly questioning whether he had the foreign policy experience and was too much of a loose cannon to be entrusted with the presidency.

Mr. Trump’s solution was to cobble together a list of men who were almost immediately written off as a collection of fringe thinkers and has-beens and unknowns in Washington foreign policy circles. Some from that group have now created far deeper problems for Mr. Trump, providing federal and congressional investigators with evidence of suspicious interactions with Russian officials and their emissaries.

Court documents and interviews with some of the advisers themselves revealed that many on the team embraced a common view: that the United States ought to seek a rapprochement with Russia after years of worsening relations during the Obama administration. Now, however, their suspected links to Russia have put them under legal scrutiny and cast a shadow over the Trump presidency.