Squire Patton Boggs, one of the largest law firms in the world, has maintained a top lobbying practice in Washington partly by bringing in lobbyists with deep connections to the leadership in both parties. But the practice, which is run by the former Senators John Breaux, a Democrat from Louisiana, and Trent Lott, a Republican from Mississippi, was not seen as particularly well connected to the Trump administration.

Before entering into a “strategic alliance” with Mr. Cohen in March last year, the firm broached the prospect of partnerships with others connected to Mr. Trump. Beginning in December 2016, Squire Patton Boggs talked with a lobbying firm started by the former Trump campaign officials Barry Bennett and Corey Lewandowski, and with David Bossie, who had served as Mr. Trump’s deputy campaign manager, among others, according to people familiar with the discussions. Unlike Mr. Cohen, the three men had extensive experience in politics and government.

Mr. Bennett, Mr. Bossie and Mr. Lewandowski, who did not end up working with the firm, declined to comment.