It comes after hospitals cancelled all non-urgent activity earlier this month amid a rising Covid-19 infection rate which was driven by the omicron variant.
The decision to pause some elective activity in the local NHS system has been "reviewed regularly" and bosses now believe it is safe to restart some services.
But Mr Burnham warned that hospitals are still under "considerable pressure".
The Labour mayor shared the news at a press conference today.
He said: “A decision has been taken across the Greater Manchester system to restart elective activity and that is happening this week.
“It has been kept under regular review and a decision has been taken now to restart activity.
“We’ll probably begin with day cases and then move to other procedures as we get into a position where that can be done.
“Patients who have had operations or procedures cancelled should just wait to hear from the NHS, but the good news is that it is starting again.
“Hopefully as we make further progress on the number of people who are in hospital who can be discharged, we’ll be able to progressively increase the level of activity and get back to what might resemble a normal level of service.
“But there is still considerable pressure on the system so people will have to bear with our NHS colleagues.
“I know it will be well received by patients who feel frustrated by the current situation and have been worried and I’m glad to be able to say that today.”
This temporary measure to pause some elective activity did not affect cancer or urgent care including cardiac surgery, vascular surgery and transplantation.
The Christie has continued to provide cancer care in a Covid-secure way and Rochdale Infirmary has continued to provide cancer and other urgent surgery.
Diagnostic services, including endoscopy, and most out-patient services have continue wherever possible.
Patients who have had appointments cancelled should wait to hear from the NHS.
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