A poll released early this month said the number of Americans who favour Mueller's probe continuing had dropped from 60% in March to 54%.
A top lawyer for Donald Trump on Sunday resumed the president's all-out attack on the investigation into possible collusion with Russia as being "illegitimate," while acknowledging a concerted effort to turn public opinion against the probe.
The comments from former New York mayor Rudy Giuliani came as Trump lashed out again at what he called "the phony Russia Collusion Witch Hunt."
For months, Trump has attacked the investigation led by special counsel Robert Mueller as politically motivated and without foundation.
His latest line of attack, which Giuliani emphasized, was the assertion that a confidential FBI informant, who met with some Trump campaign advisers in 2016 while the bureau was investigating their possible Russia contacts, was a "spy" intent on subverting the Trump campaign.
Those meetings took place during the Obama administration.
Asked on "Fox News Sunday" what was wrong with the FBI "trying to figure out what Russia was up to," Giuliani replied: "Nothing wrong with the government doing that. Everything wrong with the government spying on a candidate of the opposition party."
"That's a Watergate, spygate."
"I'm not saying Mueller is illegitimate," he said on CNN's "State of the Union." "I'm saying the basis on which he was appointed was illegitimate."
Democrats have pushed back hard at the attacks on the Mueller inquiry, which began several months after the informant's involvement. They said it had already produced real results.
Chuck Schumer, the Senate Democratic leader, tweeted: "I hate repeating myself Mr President, but let me remind you again: Special Counsel Mueller’s investigation has either indicted or secured guilty pleas from 19 people and three companies — that we know of."
And a Republican senator, Marco Rubio of Florida, told CBS that while spying on a political campaign would be wrong, "if there are people operating in this country trying to influence American politics on behalf of a foreign power, it is the FBI's job to investigate."
But what Democrats describe as a blunt and concerted effort by the president to delegitimize the Mueller inquiry may be having an effect, to judge by recent polls.
A Monmouth University poll released early this month said the number of Americans who favor Mueller's probe continuing had dropped from 60% in March to 54%.
Other polls show many Americans are unaware of the indictments and guilty pleas secured by Mueller's team.
Giuliani effectively admitted that the frequent attacks on the probe were designed to influence public opinion and take the air out of any push for impeachment.
"Of course we have to do it," he said on CNN.
"It's for public opinion," he added. "Because eventually the decision here is going to be impeach or not impeach... And so our jury - and it should be - is the American people."
Giuliani on Sunday said Trump was "adamant" about wanting to sit down with Mueller to answer questions. But, he added, "that depends on how comfortable we are with them having an open mind."
The timing of any such interview was being heavily influenced by the possible Trump summit with North Korean leader Kim Jong Un, he said.
Giuliani also said Trump had no intention of firing people to force an end to the Mueller inquiry. That, he said, would draw comparisons to President Richard Nixon's handling of the Watergate scandal.