Trump has repeatedly said there was no "collusion" with Russian Federation.
But GSA official Lenny Loewentritt said there was nothing improper about the disclosure of the emails to Mr Mueller's team.
"Not looking good, it's not looking good".
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. The president believes this, CNN's sources say, because his attorneys have tried to manage him by telling him that he faces no real danger from the probe. There's no collusion whatsoever. "A lot of lawyers thought that was pretty sad".
According to those familiar with the Trump transition organisation, the investigators did not directly request the records from Trump for America and instead obtained them from the General Services Administration.
Mueller was prompted to release a rare statement to the press, in which spokesman 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". White House attorney Ty Cobb on Saturday declined comment about the transition team's specific complaint, but he insisted to POLITICO that an ouster of Mueller isn't in the offing.
The letter has added to a recent series of statements by Trump supporters, challenging the Mueller investigation.
"Mueller will not be fired", Cobb told Fox News on Saturday. "We have continued to cooperate in every single possible way with that investigation", Mr.
Donald Trump Jr. retweeted the clip. Some reportedly came from Jared Kushner, Donald Trump's son-in-law, who is now a White House advisor.
Trump criticized the Federal Bureau of Investigation while speaking with reporters on Friday, calling the state of the agency "sad" while referring to released text messages that were critical of him between an attorney and an agent working on the case. Mueller re-assigned the agent after he found out about the critical messages. Van Grack also declined to identify the 13 transition officials whose materials the special counsel's office had obtained.
In a statement on Friday, the Justice Department Inspector General's office ― which is leading the probe of the two Federal Bureau of Investigation officials ― said it was not consulted about the release of the texts to the media, but had told officials they did not object to providing the text messages to Congress.