Drivers would pay for the courses instead of being fined and also be spared the usual three penalty points being added to their licence.
The money is required for a computer system to run the scheme.
However, Police Scotland said there were up to 30 other bids for funding from its capital budget, including body cameras for officers and new police vehicles.
Several MSPs expressed concern the scheme could be further delayed if the funding was not secured.
Chief Superintendent Garry McEwan told the Scottish Parliament's public petitions committee he was confident the money would be approved in December.
But he admitted that trying to launch the courses without the funding would be "very, very challenging".
He said: "We would have to go back to the drawing board, and look at a paper-based system."
However, the officer said the courses would still take 12-18 months to be launched, pending an organisation being appointed to run them.
In January, the Crown Office agreed in principle to introduce courses in Scotland following their widespread use south of the Border.
That came nearly three years after Police Scotland started work on the plans.
Chief Supt McEwan, who chairs a working group on the courses, said its plans had been delayed because research into courses in England was not published until August last year, a year late.
He also admitted the study for the UK Department for Transport was inconclusive on their benefits.
'Educate rather than penalise'
The officer said it had found no direct link to crash numbers, but the courses appeared to have had a "positive impact" on speeding.
He said: "It does educate rather than penalise certain driver behaviour."
Chief Supt McEwan said the working group, comprising the Crown Office, the Scottish Courts, Scottish Government and speed camera officials, was "overwhelmingly supportive" of the principle of courses.
Drivers caught speeding up to a threshold - yet to be agreed - would be offered a course instead of penalty points.
Chief Supt McEwan said that would also benefit motorists' insurance premium levels.
He said south of the Border, speeders in 30mph zones were eligible for courses if they were caught at between 39-49mph.
However, they were referred for prosecution if caught over that speed range, while no action tended to be taken below it.
The officer said the £600,000 was "not guaranteed" but he was "confident I will take receipt of the money".
But Kirkcaldy SNP MSP David Torrance was among several committee members expressing concern at the lack of progress.
He also said pressure should be put on the Scottish Government if the funding was not approved.
Mid Scotland and Fife Labour MSP Alex Rowley said the funding should be raised with justice secretary Humza Yousaf to see if money was available from other budgets.
In a statement issued later by Police Scotland, Chf Supt McEwan said: "Keeping the public safe on our roads network is a priority for Police Scotland and we are continually reviewing ways in which we can work alongside our partners to improve road safety and deter against dangerous or illegal driving practices.
"A multi-agency group has been established to examine the logistical and operational framework of offering any motorists caught speeding the opportunity to enrol in a speed awareness course.
"In addition, a course content and funding subgroup has also been established to consider possible costs appropriately.
"It is too early to speculate on the possible costs until content and course providers have been fully considered."
A Scottish Government spokesman said: “Decisions about the implementation of diversion policies in Scotland’s criminal justice system are for the Lord Advocate, in liaison with police and other partners where appropriate.
“Diversionary courses for some road traffic offences are being considered by a multi-agency working group, which will report in due course to the Lord Advocate and the chief constable.”