The top Republican Senate campaign group filed a brief on Wednesday with the United States Supreme Court in support of Donald J. Trump’s appeal of a Colorado ruling blocking him from the state’s presidential ballot. It is the latest sign of the rallying effect that the former president’s legal woes are having on his party.

The National Republican Senatorial Committee argued in its amicus brief that the Colorado Supreme Court had overstepped its jurisdiction with a decision that threatened to “unleash electoral chaos” on the 2024 presidential race.

“So even if the Colorado Supreme Court were correct that President Trump cannot take office on Inauguration Day, that court had no basis to hold that he cannot run for office,” the committee said in its court filing.

The filing from the group, the official party arm that oversees Senate races, underscores how Republicans — like the Trump campaign — are seeking to convert the former president’s court troubles into political assets. With the Iowa caucuses, the first presidential nominating contest, less than two weeks away, Mr. Trump is aiming to hold back challenges from former Gov. Nikki Haley of South Carolina and Gov. Ron DeSantis of Florida.

Primary polls show that Republican voters, who had been largely skeptical of a third presidential campaign from Mr. Trump, have coalesced around him in the last year as he was charged with 91 felonies, mostly related to his attempts to cling to power after losing the 2020 election.

Mr. Trump has been securing Republican House leadership endorsements — from Representative Steve Scalise, the majority leader, on Tuesday, and from Representative Tom Emmer, the majority whip, on Wednesday.

Senator Steve Daines, the chairman of the National Republican Senatorial Committee, is the only member of the Senate Republican leadership who has backed Mr. Trump. But the former president this week picked up a notable endorsement from Senator Tom Cotton, an Oklahoma Republican.

The committee’s amicus brief was filed by a high-profile group of lawyers, several of whom worked in the Trump administration. That list includes Noel Francisco, the former solicitor general; John Gore, who was the principal deputy assistant attorney general for the Justice Department’s Civil Rights Division; and Hashim Mooppan, a former Justice Department lawyer who defended the ex-president’s travel ban on people from Muslim-majority countries.

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