The bridge, dubbed a "blade of light" by promoters, spans the Thames between St Paul's Cathedral north of the river to the Tate Modern at Southwark on the south bank.
Engineering firm Arup is planning to attach a "shock absorber system" to the £18.2m structure, which closed in June due to excessive movement when used by crowds of pedestrians.
The bridge's operators say the engineers' solution "represents a significant step towards the reopening of the bridge".
The bridge, dubbed a "blade of light" by promoters, spans the Thames between St Paul's Cathedral north of the river to the Tate Modern at Southwark on the south bank.
It was inaugurated by the Queen in May and used by about 150,000 visitors in the first three days of opening on 10 June.
But the bridge, designed by Lord Foster, was then closed as the "synchronised footfall" effect of hundreds of people stepping in unison caused the swaying.
Arup has come up with a "damping" solution to make the bridge safe which can be compared to fitting of shock absorbers under its deck.
Viscous dampers and tuned mass dampers fitted to the structure will increase the amount of energy it can absorb and reduce the sway.
Unwanted vibrations
Both types of devices have been used in other constructions and mechanical systems to prevent unwanted vibrations.
Operators had considered stiffening the bridge by adding new elements and also limiting the number of people on the structure, but both were considered unsuitable solutions.
Testing the dampers on the bridge will take approximately six weeks.
If successful, operators hope to start the full remedial work which is expected to take a further six months.
When the bridge was closed, architect Lord Foster said he was bitterly disappointed, but insisted the problem was not a safety issue and would not have chosen any other design.
The completed bridge, owned by the London Borough of Southwark, is already reported to have been £2m over budget.
Extra costs
The agreement for the new tests has been made by the Millennium Bridge Trust, Southwark Council and the Corporation of London.
As well as the immediate multi-million pound repair costs, additional costs of long-term maintenance are expected.
The bridge operators said: "Everyone involved with the bridge is committed to a permanent opening at the earliest possible date."
The bridge has received £7.1m funding from the Millennium Commission and other financial backers have included the Corporation of London and the London-based Cross River Partnership.
Organisers had hoped that four million people would cross the bridge in its first year of opening.