Senator-elect Doug Jones, D-Alabama, told "State of the Union" that he believes the investigation will roll on, but added that "anybody would need to be concerned if Bob Mueller was sacked".
The investigation involves whether the Trump campaign helped out in Kremlin-directed cyberhacking efforts to helped the president win the election.
At issue is how the special counsel obtained the materials. It's a government agency that does a range of things like providing office space for federal employees.
The emails in question involve 13 transition officials, including four senior ones, according to the letter.
The special counsel's office pushed back on the characterizations that the emails were obtained unlawfully.
Asked if Mr. Trump was "setting the stage for firing Bob Mueller", Mr.
Allen has posted a PDF of the seven-page letter dated December 16 from transition team lawyer Kory Langhoffer to Senator Johnson et al.
Langhofer wrote that Mueller's office "received from the GSA tens of thousands of emails, including a very significant volume of privileged material", according to a copy of the letter published by Politico.
Mr Langhofer said a GSA official appointed by Mr Trump in May had assured the transition in June that any request for records from Mr Mueller's office would be referred to the transition's lawyers. Officials with Trump for America learned last week that GSA officials had turned over the cache of emails to Mueller's team.
Putin: Trump's call saved Russian Federation from deadly terror attacks this weekend
This is the second time in less than a week that two of the world's most powerful leaders have exchanged messages of praise. Petersburg, allowing Russian police to arrest a group of suspects in raids on Friday.
"Our committee has direct jurisdiction over the Presidential Transition Act, and it simply does not support withholding transition team emails from criminal investigators", the Maryland Democrat said.
Complicating matters is that Langhofer, the lawyer who sent the letter to the Congressional committee chairs, said in the memo that there was an understanding that TFA was the custodian of the emails.
Mr Beckler was hospitalised in August and has since died.
After returning from Camp David, President Trump says he's not planning to fire special counsel Robert Mueller.
Speculation about whether Trump will fire Mueller started after Rep. Jackie Speier, D-Calif., said in a local TV interview Friday that "the rumor on the Hill" was Trump was going to fire Mueller at the end of this week, after members had left Washington for Christmas break.
If Trump knows he has something to hide that's potentially impeachable, he should probably - putting ethics and the truth aside - fire Mueller and try to out-run the law. "We've continued to cooperate at every single possible way with that investigation".
I have to say that there would be something flawless about this - a special counsel investigating, in part, the circumstances that led to his appointment would be fired, perhaps leading to an impeachment about the circumstances of his ouster.
As it stands, Mueller's team has secured indictments against Trump's former campaign chairman Paul Manafort and Manafort's business associate Rick Gates.