Special counsel Robert Mueller has reportedly obtained tens of thousands of emails from President Donald Trump's transition team as part of a probe into the 2016 election.
So far the FBI has charged four people as part of the investigation.
A lawyer from the Trump for America group has alleged the emails were obtained unlawfully from a third party.
But a spokesperson for the special counsel said the "appropriate criminal process" had been followed.
Mr Mueller is investigating alleged collusion between the Trump presidential campaign and Russia, something both deny.
Trump for America used a government agency, the General Services Administration (GSA), for their offices, equipment and email hosting in the period between Donald Trump's election and his inauguration in 2017.
The GSA is reported to have supplied these records to Mueller's investigation team in the summer.
Kory Langhofer, a lawyer for Trump for America, sent a letter to Congress on Saturday complaining that the GSA "did not own or control the records in question", alleging US lawmakers' constitutional rights may have been breached by the disclosure.
The letter says the group believe some of these communications should have been redacted because they believe they contain information under privilege. The seven-page letter was published by political news website Politico.
Mr Langhofer said Trump for America had separated those emails out in advance of any request to share them, but found out this week that Mr Mueller's probe had already obtained the emails months ago.
The transition group lawyer is requesting that Congress act to protect future presidential transitions from having "private records misappropriated by government agencies, particularly in the context of sensitive investigations intersecting with political motives".
But a spokesperson for Mr Mueller, Peter Carr, said: "When we have obtained emails in the course of our ongoing criminal investigation, we have secured either the account owner's consent or appropriate criminal process".
Others too have defended against the accusations the emails were obtained unlawfully. Democratic Representative Eric Swalwell tweeted that the claims were "another attempt to discredit Mueller as his #TrumpRussia probe tightens"".
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American news website Axios reported on Saturday that email records from 12 members of the campaign, including President Trump's son-in-law Jared Kushner, had been obtained by the Mueller team.
President Trump's private lawyers are expected to meet Mr Mueller and members of his team next week to discuss the next phases of the investigation, US media report.
Earlier this month the president's former national security advisor Michael Flynn became the highest-profile person charged in the probe, pleading guilty to making false statements to the FBI about meetings with Russia's ambassador weeks before Donald Trump's inauguration. It was revealed he is co-operating with the inquiry's investigation.
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