Site sources told the Enquirer that the digger landed on a cherry-picker at the bottom of the shaft.
One said: “The two injured were working on the cherry picker down the shaft which was a write-off.
“They were lucky not to be killed in an accident like that.”
The Greenwich site is part of the Tideway east section where a joint venture of Costain, Vinci and Bachy Soletanche are main contractors.
A Tideway spokesperson said: “We can confirm that there was an incident during a lifting operation at our site in Greenwich on the evening of 22 May.
“Two members of our team were taken to hospital and have since been discharged. Fortunately there were no serious injuries.
“Safety is our top priority and we are now investigating the cause of the incident.”