The Government will keep a "close eye" on Kensington and Chelsea council, Sajid Javid said, as pressure built for the troubled authority to be taken over by commissioners.
The communities secretary's intervention followed resignations by the council leader and deputy leader under intense criticism of how the Grenfell Tower tragedy was handled.
Mayor of London Sadiq Khan earlier led calls for commissioners to be drafted in to take over the day-to-day running of affairs in the west London borough.
Mr Javid said: "It is right the council leader stepped down given the initial response to the Grenfell tragedy.
"The process to select his successor will be independent of government, but we will be keeping a close eye on the situation. If we need to take further action, we won't hesitate to do so."
Outgoing leader Nicholas Paget-Brown said he had to share responsibility for "perceived failings", departing alongside deputy Rock Feilding-Mellen.
The council has now been hit by three high-profile resignations in the aftermath of the fire, with chief executive Nicholas Holgate also exiting.
The authority came under fire for its slow response to the disaster, in which at least 80 people are thought to have died.
The Labour Party launched a drive for the council to relinquish its handling of affairs until the crisis was brought under control, with a string of party figures weighing in.
Mr Khan, who welcomed the resignation, said the Government had "no option" but to appoint "untainted" commissioners who had "a genuine empathy for local people and the situation they face" to take over running the authority.
This woman gave a passionate speech to MPs about what she says are the failings in how the Grenfell Tower victims are being treated. pic.twitter.com/s1PXDe4gOL— Channel 4 News (@Channel4News) July 1, 2017
He said: "The council now needs to find a way to move forward and find a way to restore the confidence in that community.
"That can only be done with new leadership and a new approach that reaches out to residents who quite rightly feel desperately neglected.
"I ... feel the response from the council and subsequent breakdown in trust is so severe that there is now no alternative and the Government needs to step in quickly."
Commissioners were brought in to run Tower Hamlets Council in 2014 after a critical independent report into the council's award of grants and sale of properties under former mayor Lutfur Rahman.
Shadow housing secretary John Healey said the move would amount to "specific, immediate, obvious and necessary action", while shadow communities secretary Andrew Gwynne added the powers should be used to "get a grip on what has gone dreadfully wrong".
Meanwhile, Yvette Williams, one of the organisers of the Justice 4 Grenfell campaign group, said the community would not accept commissioners imposed on them without consultation.
She said Grenfell survivors and victims' families are "very, very angry" and claimed that one former resident called her on Friday to reveal their rent for the tower block had been deducted from their bank account.
'Nothing is off the table' if the Grenfell Tower council fails to care for victims, says Communities Secretary amid calls for a takeover. pic.twitter.com/CVpzGdH4WV— ITV News (@itvnews) July 1, 2017
It comes as Jeremy Corbyn said he has written to Prime Minister Theresa May urging her to widen the scope of the public inquiry.
He has asked for a two-part inquiry, the first looking at specific issues around the fire in at the 24-storey building in Kensington, west London, and reporting back soon, with an additional second part "looking at the national issues".
In his resignation statement, Mr Paget-Brown acknowledged the council had been criticised for "failing to answer all the questions that people have" but that the scale of the tragedy "was always going to mean that one borough alone would never have sufficient resources to respond to all the needs of the survivors and those made homeless, on its own".
Speaking later from near the Grenfell Tower, Mr Javid reiterated no options were off the table in the Government's approach to the council.
He told the BBC: "In terms of any kind of intervention for any council, whenever these kind of decisions are made and they are very infrequent - and rightly so because we would be overturning your democratically elected politicians - they are quasi-judicial decisions.
"It wouldn't be appropriate for me to talk about any particular council or any particular decision that may or may not happen.
"But one thing is clear, the absolute priority remains looking after the victims, their families and friends, making sure they get everything they need and, in doing so, when it comes to the local council, nothing is off the table."
Labour councillor Robert Atkinson, who was locked in a heated confrontation with Mr Paget-Brown at a cabinet meeting on Thursday, agreed action was needed.
He told BBC Radio 4's PM: "The council has failed in its obligation and I don't see how they can recover the confidence of the residents with the current administration.
"I do not think this is the time for constitutional bickering between us, we do need to act, we don't have time to wait for this.
"I would give the Tories until Monday to find a leader and a leader who has got the confidence to get to North Kensington, who will speak to the people of North Kensington and who makes sure people are moved out of hotels, that those who are in need in the area who still have no heating get their heating."
He added some of those displaced by the fire were now sharing hotels with tennis fans arriving to watch Wimbledon next week.
Asked if he would have been willing to form an emergency coalition with the Conservatives, he said: "We went to that disastrous meeting with the intention of offering our support and feeding information and it wasn't wanted and that is exactly what has been the problem all along."