A source told the Local Democracy Reporting Service this was planned as part of the national vaccination programme
In Nottinghamshire, health bosses say they will have to lower the number of jabs given over the next 10 days as a result of "reduced supply over the next week".
The CCG for the area said it anticipates supplies will then "increase significantly" over the next few weeks, meaning more appointments will be made available.
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A CCG spokesperson said: "Fluctuations in vaccine supply has been planned for and factored into the planning of the vaccine rollout.
"We would advise that if there is no availability at the site you want to visit, you should keep checking back each day as new appointments are released on a regular basis."
In Derby and Derbyshire the CCG said it was preparing to receive an increase in vaccines "within the next two weeks".
It did not say if supplies would be limited next week but said some first dose appointments were not available due to staff giving second doses and vaccinating housebound patients.
Public Health England has reported on Twitter that as of Saturday, March 6, 2021, there were 6,040 new cases and 158 deaths within 28 days of a positive test reported across the UK. A total of 21,796,278 people have now received the first dose of a #vaccine.
Its COVID19 dashboard has also been updated and the daily figures can be seen at http://coronavirus.data.org.uk