Around 5,000 people were evacuated when the blaze broke out under a pavement in Holborn on 1 April.
Since then work has been underway to repair the damage caused to a number of road tunnels.
The Strand Underpass will be the first to reopen to traffic. From 12 June buses and taxis will be allowed to use the northbound tunnel of the Kingsway.
All the roads affected should be fully reopened within the next few weeks.
Workers have been operating in confined spaces to shore up the tunnels with wood and repair the brickwork.
London Fire Brigade said investigations were still going on into the exact cause of the fire, which took 36 hours to extinguish.
Fire crews were called when smoke was seen coming out of an inspection cover on Kingsway.
Firefighters worked day and night to put the fire out and stop it spreading to buildings in the area.
Several performances at the Lyceum, Duchess and Aldwych theatres were cancelled and 1,900 customers in the area were left without electricity.
A major highway which has been closed since an underground fire in central London will reopen on Wednesday.