Under the plans flu jabs would also be compulsory for those in contact with patients and people receiving care.
Health Secretary Sajid Javid has urged all health service workers to get both jabs to protect vulnerable patients.
The government says 88% of NHS trust staff have received two Covid doses.
If the proposals go ahead it would mean that only those who are fully vaccinated, unless medically exempt, would be able to be deployed to deliver health and care services.
From 11 November it will already be mandatory for workers in care homes in England regulated by the Care Quality Commission (CQC) to be double jabbed unless they are exempt.
MPs backed the change in July but a number of Conservative rebels voiced anger at the plans.
Care organisations also warned compulsory vaccinations could cause difficulties in a sector that already struggles with recruitment.
The Scientific Advisory Group for Emergencies (Sage) Social Care Working Group has advised that the overlap between the sectors makes a strong scientific case for there to be similar approaches to vaccination.
The Department of Health and Social Care (DHSC) said the consultation would focus on the proposals, their scope, and any potential impact mandating vaccines could have on staffing and safety such as reducing staff sickness absence.
Staff, healthcare providers, stakeholders, patients and their families are all being urged to share their view with government with a final decision is expected this winter.
Mr Javid said it was clear to see the impact vaccine shad against respiratory viruses which could be fatal to the vulnerable.
"We will consider the responses to the consultation carefully but, whatever happens, I urge the small minority of NHS staff who have not yet been jabbed to consider getting vaccinated - for their own health as well as those around them," he said.
The DHSC said that there is a longstanding precedent for vaccination requirements in NHS roles with policies already in place requiring the Hepatitis B jab for those undertaking procedures prone to exposure to the virus - such as surgeons.
Northern Ireland, Scotland and Wales have not made Covid vaccines compulsory for care home staff.
In Scotland MSPs are to vote later on plans to introduce a vaccine passport scheme for people attending nightclubs, major sporting events and many concerts and festivals - a similar scheme is planned for England from the end of the month.
Flu vaccination has also been recommended to staff and vulnerable groups in the UK since the 1960s.
National flu jab rates in the health service increased from 14% in 2002 to 76% last year.
The Joint Committee of Vaccination and Immunisation (JCVI) has warned there could be a surge in flu cases this year due to coronavirus lockdowns meaning flu levels had been very low last winter.
One of the largest suppliers of seasonal flu vaccines has said there have been delays to the rollout in England and Wales due to a shortage of HGV drivers.
On Wednesday Boris Johnson said the number of unvaccinated people ending up in hospital with Covid was "concerning".
The UK has reported more than 30,000 new daily cases for the last nine days in a row and the number of people in hospital with the disease is also rising.