Greg Hands has refused to deny putting pressure on the leader of Kensington and Chelsea Council to resign.
The Chelsea and Fulham MP said he would not comment on whether Whitehall had pushed for the departure of Nicholas Paget-Brown last week.
The council leader resigned amid a storm of criticism about his response to the deadly fire.
Speaking to the BBC's Sunday Politics, Mr Hands claimed it was "natural" to have discussions with council leaders as an MP.
Pressed by host Andrew Neil on whether he leaned on Mr Paget-Brown, he replied: "Well I'm not going to comment on that as a local Member of Parliament.
"As a local Member of Parliament, I speak to my council leader frequently, as I think all local MPs do."
"So you were involved, you nudged him to go?," Mr Neil replied.
"Well I talked to that leader and the previous leader and the leader of my other council. It's natural that the MPs speak to their local council leader on an ongoing basis," the minister said.
Addressing claims that local authorities had been starved of funds, leading to cuts in areas such as fire safety, Mr Hands pointed to the £8.6 million found by the council to finance the refurbishment of Grenfell Tower.
"There's not necessarily a sign of a shortage of funds in doing that," he said.
