This would include raising the required standard of entry and setting up a scheme - called Teach Now - to encourage people who had succeeded in other professions to go into education.
To cope with a shortage of maths and science teachers, the Conservatives are promising to pay off student loans for applicants with upper second- or first-class degrees in these subjects from "good universities".
Meanwhile, financial help with postgraduate teacher training would be removed for those who have achieved a third-class degree or lower.
But, according to the Training and Development Agency for Schools in England, this only accounts for 2% of postgraduate primary trainees and 4% of postgraduate secondary trainees.
Unveiling his proposals, Mr Cameron told an audience at a school in south-east London he would make teaching "the new noble profession".
He said: "We need much greater flexibility than currently exists - flexibility over rewarding the best and, yes, getting rid of the worst.
"So we will free schools to pay good teachers more. With our plans, head teachers will have the power to use their budgets to pay bonuses to the best teachers."
Mr Cameron said: "Everyone remembers a teacher who made a difference through sheer force of personality."
He added: "The quality of a teacher is the single most important factor in a child's educational progress.
"Those taught by the best teachers make three times as much progress as those taught by the worst."
He highlighted the education systems in Finland, Singapore and South Korea as deliberately promoting teaching as a prestigious profession.
"They are brazenly elitist - making sure only the top graduates can apply. They have turned it into the career path if you've got a good degree.
"And in America, President Obama is offering financial incentives to attract more science graduates into teaching. We should be equally bold here."
Mr Cameron said the existing Teach First scheme - which aims to get "exceptional graduates" working in "challenging" secondary schools - had to be widened to include institutions placed in "special measures". This is currently not permitted.
For Labour, Schools Secretary Ed Balls said: "The fact is that teaching has been transformed from a demoralised profession in 1997 to the number one choice for graduates today.
"To attract top professionals to make a career change into teaching we are already working with over 400 leading employers, focusing on key subjects like maths and science.
"And to put teaching on the same footing as high-status professions like doctors and lawyers we are introducing a new Licence to Practise with a right for all teachers to get ongoing training and career development. It's time the Tories backed these reforms."
Liberal Democrat schools spokesman David Laws said: "With the Conservatives targeting the education budget for cuts, their proposals will mean fewer teachers and even larger class sizes.
"We need to get better qualified people into teaching, but the Tories can't be trusted to do this. Their plans to slash education spending will mean that schools will simply not have the cash to pay good teachers more."
Christine Blower, general secretary of the National Union of Teachers, said: "Being 'brazenly elitist' could mean being brazenly exclusive of those potential teachers who through no fault of their own have had a tough time in achieving the necessary qualifications.
"Teaching is an extremely demanding profession and not everyone can do it, even those with first-class degrees.
"While qualifications are obviously necessary, being a good teacher is not dependent on academic ability alone."
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