Senior White House adviser Kellyanne Conway refused on Sunday to say if Donald Trump will commit to not firing Robert Mueller, the special counsel investigating links between Trump aides and Russia who has empanelled a grand jury in Washington.

One of four senators who have introduced legislation to protect the former FBI director, meanwhile, said his firing “would be crossing a big line” and would likely see his immediate reinstatement by Congress.

Conway appeared on ABC’s This Week. “The president has not even discussed that,” she said. “He’s not discussing firing Bob Mueller. We are complying and cooperating ...”

Host George Stephanopoulos then interjected: “But will he commit not to fire him?”

“He’s not discussed firing Bob Mueller,” Conway said.

“That’s not what I’m asking,” Stephanopoulos said.

“Hold on,” said Conway, “I’m not the president’s lawyer here. But I will tell you as his counsellor he is not discussing that.

“You have to listen to his special counsel Ty Cobb who works in the White House now, and he said very clear this week that we will continue to cooperate with Bob Mueller and his investigation even though … many of [Mueller’s team] are Democratic donors, but we’ll continue to co-operate and comply.”

In June a close associate of Trump, Newsmax chief executive Chris Ruddy, told PBS the president was considering firing Mueller.

Conway also said investigations into Trump and Russia were “a completely false and fabricated lie” and said: “But so far you’ve got … no collusion. And anybody who denies that is lying.”

On Thursday night in West Virginia, Trump disparaged Mueller’s work, telling supporters the Russia investigations were part of an effort “to cheat you out of the leadership that you want with a fake story”.

Among other matters, Mueller is investigating a June 2016 meeting at Trump Tower in New York City between Donald Trump Jr, Trump’s son-in-law and adviser Jared Kushner, then campaign manager Paul Manafort and a Russian lawyer. Other Russians were also in the room, one of them with ties to Soviet counter-intelligence.

According to emails he published himself, Trump Jr believed the lawyer would offer damaging material about Hillary Clinton, his father’s opponent in an election US intelligence agencies agree Russia sought to influence on Trump’s behalf. Kushner and Manafort have denied collusion.

On Sunday, Trump surrogate and governor of New Jersey Chris Christie told CNN’s State of the Union the meeting was “ill-advised”. It was not clear if Trump himself had known about the meeting at the time, Christie said.

White House press secretary Sarah Huckabee Sanders admitted this week that the president played a role in drafting the first and misleading official statement about the meeting, “as any father would”. That contradicted another of the president’s lawyers, Jay Sekulow, who said Trump had not been involved in the statement.

The existence of the Washington grand jury was reported by the Wall Street Journal on Thursday. Most observers agreed it was bad news for Trump, who could be called to give evidence. The panel, formed in a heavily Democratic city, is empowered to look into the Republican president’s financial history.

Mueller’s investigative team, which is considering whether Trump obstructed justice when he fired FBI director James Comey in May because of “this Russia thing”, is staffed with experts in financial crime.

Christie, a former US attorney, told CNN a grand jury was “a normal step taken by a careful prosecutor who is doing a thorough investigation. And I think that’s exactly what Bob Mueller is doing. You can’t issue subpoenas without a grand jury. It’s the grand jury that actually issues the subpoenas.”

On Friday the New York Times reported that Mueller’s team asked the White House for documents related to Michael Flynn, the retired general who resigned as Trump’s first national security adviser after he was found to have misled Vice-President Mike Pence over meetings with the then Russian ambassador.

Before the Senate left Washington for its summer recess this week, two pairs of senators introduced legislation to protect Mueller from being fired by Trump. One of those senators, the Delaware Democrat Chris Coons, told ABC on Sunday: “I think if the president should fire Robert Mueller abruptly, that would be crossing a big line.

“I think you would see strong bipartisan action from the Senate, which might include our reinstating him or our hiring him to continue to conduct that investigation on behalf of Congress.”

Speaking to CBS’s Face the Nation, the Arkansas Republican senator Tom Cotton said he did not think such legislation would “go very far”.

But he indicated a willingness to stand up to the Trump White House, adding: “For decades, Congress has ceded too much authority to the executive branch. And we should exercise our constitutional responsibilities seriously and with vigor.”

In the Philippines, meanwhile, Trump’s secretary of state, Rex Tillerson, had his first meeting with Russian foreign minister Sergey Lavrov since the president signed – and criticised – new sanctions on Russia.

Despite the Putin regime retaliating with measures against US diplomats in Russia, Lavrov told reporters he “felt the readiness of our US colleagues to continue dialogue”.

“I think there’s no alternative to that,” he said.