Holland-Kaye confirmed on Thursday (11 June) the hub had agreed a severance scheme with its unions over an unspecified number of job losses.
He also declined to rule out further job cuts, but stressed the airport would seek to minimise job losses wherever possible.
It comes after passenger numbers fell to an all-time low in May, down 97% year-on-year.
"In line with this decline, the airport has begun to restructure its frontline roles, having already cut a third of managerial roles," said Heathrow.
The "grim" situation is set to be compounded in June by the government’s 14-day quarantine on arrival policy, introduced on Monday (8 June).
Heathrow has backed calls for ministers to urgently establish air bridge or safe travel corridor arrangements with countries with lower rates of coronavirus infection to restart flying this summer.
"Throughout this crisis, we have tried to protect front line jobs, but this is no longer sustainable, and we have now agreed a voluntary severance scheme with our union partners," said Holland-Kaye.
"While we cannot rule out further job reductions, we will continue to explore options to minimise the number of job losses.”
The airport is also supporting calls for airports in England and Wales to receive a 12-month business rates waiver, mirroring support extended to airports in Scotland and Northern Ireland.
Last month, Heathrow began trialling thermal screening technology to examine how such measures could be rolled out more widely to reduce coronavirus transmission, and how they could form part of a common international standard for airport health screening.