There will be no replacement buses but paper tickets will be accepted on routes 54, 380 and SL3.
Taxis will be provided for passengers with accessible needs for travel between Blackheath and Charlton.
Mr Davidson added: “We know there’s never a good time to close the railway, and that changes to services for a 10-week period is a long time.
“These essential repairs will improve the safety and operation of the tunnel.
“If traditional weekend working was used engineers would spend at least 50% of a weekend bringing materials and plant in and out of the tunnel, leaving limited time for actual work.”