The Liberal National Party would be ready for a byelection in the far north Queensland seat of Cook that could see it take power in Queensland, the Opposition Leader said on Saturday.

But Lawrence Springborg said embattled Labor MP Billy Gordon, who has had serious allegations levelled against him, was still entitled to remain in his role as the member for Cook "until he is no longer considered to be the member for Cook".

The stakes could not be much higher, with the new Labor government just one seat away from opposition.

Mr Springborg said Premier Annastacia Palaszczuk should have acted more quickly when she learnt of the allegations made against Mr Gordon.

"There are some very serious questions here, given that there is a litany and legacy of issues around the non-paying of child support, the non-filing of tax returns and also this issue of serious allegations of abuse," he said.

"It's up to the Premier, who set these very high standards, to indicate today whether he fulfils those standards."

An "absolutely furious" Ms Palaszczuk referred Mr Gordon to police on Friday evening, when allegations against Mr Gordon by a former partner became public.

Mr Springborg said the Premier had "failed her first test of accountability" when she did not refer the matter to police on March 18, when Mr Gordon's former partner wrote to Labor and LNP MPs.

"If it's good enough to refer to police yesterday afternoon when the media found out about it, it was good enough to refer it to police on the 18th of March when the Premier received that letter," he said.

"With regards to the former Labor candidate in Lytton [Daniel Cheverton], the Premier intervened immediately and said [Mr Cheverton] did not fulfil her very high standards and disendorsed that candidate.

"I just want to know what the difference is between the member for Cook and the former Labor candidate for Lytton."

Mr Cheverton was stripped of his preselection late last year amid claims of "inappropriate" behaviour towards a female colleague.

Mr Springborg said the LNP would be ready for a byelection if Mr Gordon's position as a member of Parliament became untenable.

"The LNP will always deal with every eventuality," he said.

Comment was sought from Ms Palaszczuk.

In his personal address to Parliament on Friday morning, Mr Gordon said his relationship with his former partner "included good times and not so good times".

"There were times when both of us failed to behave as adults should and I certainly bear full responsibility for my actions," he said.

"Unfortunately the relationship did not end on a positive note.

"It was an extremely upsetting period, after which I independently sought counselling.

"The counselling involved the self-appraisal process where I admitted to my former partner some personal failures as well as challenges I had faced in my life.

"Over time, I moved on and feel privileged to have been able to start afresh while at the same time providing support to my ex-partner and my children."

The allegations against Mr Gordon were published on a website run by former journalist and Cairns LNP MP Gavin King.