The offers would be made to members of the 1,400-strong civilian staff, although targets for the number of job reductions which may be sought, or in which areas, have not been made.
Police chiefs said the scheme was only being established "in principle", and would only be activated if budget shortfalls meant savings needed to be made.
The proposed scheme forms part of a new review of the force entitled "Transforming the Service" (TTS), led by Deputy Chief Constable Steve Allen.
The TTS programme will scrutinise many areas where savings can be made, including cutting police overtime, which totalled 8.5 million for 2009/10.
But with 85 per cent of the force's 172.6m budget for the coming year going on salaries, police chiefs are set to establish the voluntary scheme in an effort to avoid future compulsory cuts.
Deals would only be offered to civilian staff as police officers cannot be made redundant by law.
In a report to the police board, Chief Constable David Strang said the force was facing a funding reduction of 12 per cent in real terms over the next three years, while some predictions put the figure at 20 per cent over five years.
Mr Strang said: "It is clear that the force must plan now for actions that must be taken both in the short-term and in the coming years."
DCC Allen added: "At the police board next week, we are asking the board to approve in principle a voluntary redundancy scheme. No decisions have been made but we would like to be in a position where we know we could activate a scheme for volunteers if the need to reduce numbers arises.
"All these measures are about making sure we do not make staffing decisions now that make it more difficult to achieve savings later on."
DCC Allen said trade unions will be consulted.
Under the scheme, police staff accepted for voluntary redundancy or early retirement could leave as early as April next year.
Councillor Iain Whyte, convener of the police board, said: "We've been told to plan on at least a 4 per cent reduction, or 12 per cent in real terms, for the next three years, and possibly longer than that. For an organisation where 85 per cent of the costs relate to staff salaries, it's entirely sensible that a system is put in place where voluntary redundancies and early retirements can be put in place if that helps."
Jackie Muller, secretary of the Lothian and Borders branch of the Scottish Police Federation, said: "Any reduction amongst support staff must not impact on the number of frontline police officers."
Lothian and Borders Police is funded by payments from the five local authorities covered by the force and the Scottish Government.
A city council spokesman said: "We recognise the importance of policing but also have to take into account all the other services we provide to our communities."
A Scottish Government spokeswoman said: "All operational decisions are a matter for individual chief constables."