Bizjak Mlakar has been under fire for weeks but indicated she will not step down, which leaves the prime minister no other choice but to seek a dismissal motion in parliament.

The meeting with the prime minister came amid growing signs that Bizjak Mlakar has run out of favour with Cerar as well as her Pensioners' Party (DeSUS) over the management of the UNESCO-listed mercury mine in Idrija.

She has refused to implement a government decree bringing the closed Idrija mine under the management of the Culture Ministry, arguing that this is the purview of the economy portfolio.

But Bizjak Mlakar said today she only wanted to prevent danger to people and property that could ensue if the Culture Ministry was to take over the Idrija mine.

"Women who think with our own head and refuse to listen to men have to be subjugated," she told reporters after the meeting.

She later elaborated that Cerar had asked her to resign as reported by some media, but stressed she would not heed his call.

While the Idrija mine is the ostensible cause of her demise, Bizjak Mlakar has been the subject of heavy criticism throughout her tenure, with complaints coming from journalists and media owners over media legislation, and numerous artists over funding and culture policy in general.

But she had successfully deflected all appeals for her dismissal, which commentators have ascribed to her powerful position in the DeSUS, the coalition kingmaker.

It was not until DeSUS president Karl Erjavec last week said her fate was in the prime minister's hands that the opening was created for her departure.

Erjavec said today that he had supported the minister but in this case it was "about Cerar and the minister. I tried to help but I failed, they are simply too wide apart."

Given her refusal to resign, it appears likely Cerar will formally ask the National Assembly to dismiss her. A simple majority is enough to fire a minister.

Erjavec would not say today whether DeSUS would vote in favour of her dismissal. Bizjak Mlakar did not specify whether she expected DeSUS MPs to support her.

If dismissed, Bizjak Mlakar would become the sixth minister in the Miro Cerar cabinet to be replaced.

Janko Veber was sacked as defence minister in a similar way: having refused to heed calls for dismissal, he was voted out of office in April 2015.

Bizjak Mlakar, a mathematics professor with an MBA degree, spent most of her career in education and local politics, having also been active in DeSUS.

She served as MP for DeSUS in 2009-2011 and was re-elected to parliament in 2014.

According to law, Bizjak Mlakar can return as MP if she is dismissed. She told reporters she would return to parliament but her return to the DeSUS deputy group would be "the subject of discussion".