During an interview with ABC News, Sekulow said, "The President has the authority to take action".
"I think the best case for the President is to be vindicated by allowing this investigation to go on thoroughly and independently".
Mueller is a former FBI Director who unlike Trump understands how Washington, D.C works. In what became known as the Saturday Night Massacre, President Richard Nixon ordered then-Attorney General Elliot Richardson to fire Cox.
And that boss, Jeff Sessions, also promised Tuesday not to get involved in any firing.
"I'm not going to follow any orders unless I believe those are lawful and appropriate orders", Rosenstein said.
So that would leave such a decision to Rosenstein, who just appointed Mueller on May 17 to oversee the federal investigation into Russian interference in the 2016 election.
As special counsel Robert Mueller builds his legal team, Mr Trump's allies have begun raising questions about the former Federal Bureau of Investigation director's impartiality, implying he can not be trusted to lead the probe.
Rosenstein was likely referring to the possibility that Mueller would investigate whether Sessions violated his recusal or whether Comey's firing was an effort to obstruct justice.
High-profile supporters of Donald Trump are turning on the man investigating Russian interference in the U.S. election and possible collusion with the president's campaign - with one suggesting Mr Trump was considering "terminating" him.
As Mueller builds his legal team, Trump's allies have begun raising questions about the former Federal Bureau of Investigation director's impartiality, suggesting he can not be trusted to lead the probe.
Knives coming out: Trump fans float the idea of firing investigator Mueller
"I think he is considering perhaps terminating the special counsel", Ruddy said. "I think there's a conflict there". Comey, however, specifically said at congressional hearing that Mueller had not reviewed or edited his testimony.
During the meeting the president reportedly told Comey that he hoped the Bureau would not pursue the investigation into former White House National Security Adviser Michael Flynn.
Ruddy, who is a friend of the president's, said on PBS Monday, "I think [Trump is] considering perhaps terminating the special counsel..." He thinks Trump is "weighing that option".
Jeanne Shaheen, a New Hampshire Democrat, whether he has seen good cause to fire Mueller.
Trump has repeatedly denied colluding with Russian Federation during the presidential campaign, and Comey and others have testified that there is no evidence to the contrary.
Chris Ruddy to @JudyWoodruff: President Trump is considering firing special counsel Robert Mueller, who he considered for another position.
Still, Spicer's remarks irked Ruddy, who says he should not have been asked to release a statement saying that he never spoke to the president because he never claimed to have spoken with the president about Mueller. He said he would agree to dismiss Mueller only if there were a legitimate basis to do so. The person demanded anonymity to discuss strategy on the sensitive matter.
Rosenstein's letter critical of Comey's handling of the Hillary Clinton email server investigation served as the Trump White House's first reasoning for firing Comey - though Trump later undercut that explanation by saying he fired Comey in hopes of ending the Russian Federation probe.
"Look, the president of the United States, as we all know, is a unitary executive".
"It's a mistake to pretend that this is going to be some neutral investigation", Gingrich said. "Chris speaks for himself", the official said.
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