Storms Desmond, Eva, Frank and Gertrude all caused major issues in the area, as did poor weather on 27 January.
A report to councillors said nearly 400 repair jobs eligible for support under the Bellwin Scheme had been identified.
Their cost is likely to exceed £3.5m. Scottish Borders Council is bidding for a further £925,000 from another scheme.
The Scottish Borders was hit by heavy rainfalls throughout the winter with Hawick, Jedburgh, Newcastleton and Peebles badly affected.
A lengthy list of repairs has been drawn up across the region and many of them have already been carried out.
Most of them should be funded by the Scottish government via the Bellwin Scheme.
However, SBC will need to fund the first £508,000 of works.
The local authority also hopes to get further capital funding from a separate government-backed compensation scheme.
It would be used to replace the Bowanhill Bridge near Teviothead and Newmill footbridge north of Jedburgh, and carry out embankment works on the Ettrick Water in Selkirk.
The latest estimate of the cost of repairing the damage caused by severe winter weather in the Borders has said it could exceed £3.5m.