It comes after many Scots in the 60 to 65 age group took to social media during the week say they had not received a letter, but on calling the national vaccine helpline had been told they had an appointment already scheduled which they had not been notified of.
Many of those raising the issue were based in Edinburgh and the Lothians.
Asked about the issue at the Covid-19 daily briefing on Friday, health secretary Jeane Freeman said she was not aware of any issues in the Scottish Government system for distributing letters.
But NHS Lothian later confirmed that some letters had been delayed, adding that the issue should be resolved by the weekend.
"Appointment letters are being distributed by a central system to patients across Scotland,” a spokesperson said.
"We understand there has been a delay in sending out some appointment letters this week (March 12).
"This should be resolved very quickly and the letters are expected to arrive by this weekend. However, if you are over 60 and have not received an appointment letter by Monday (March 15), please call the national helpline on 0800 030 8013.”
Labour MP for Edinburgh South Ian Murray said he had seen evidence of a “significant problem” with vaccination letters in Edinburgh.
"Yet again the Scottish Government denies there is a problem, when the reality on the ground is much different,” he said.
“Many people have only learned of their appointments by calling the helpline and that is simply not good enough. It could go some way to explaining why Edinburgh is behind every other area in Scotland.
“Rather than denials, ministers need to get a grip. It’s too important to get wrong.”
Conservative Edinburgh city centre councillor Jo Mowat wrote on Twitter: “Hearing of problems with letters being sent out with appointments for vaccination in Lothian – if you are aged between 60-65, due for Covid-19 vaccine, live in Lothian not received an appointment letter then phone 0800 030 8013.”
Asked about the reported problems with appointment letters during the coronavirus daily briefing on Friday, health secretary Jeane Freeman said she was not aware of any issue.
“At this point we've not identified any system problems, but we will continue to look and make sure that the vaccine management tool, and the booking system are working well together, and that we're getting that information out to people as quickly as we possibly can,” she said.
A Royal Mail spokesperson denied any fault lay with them.
They said: “Letters informing people of upcoming vaccination appointments are being sent in good time and according to plan.
"We have been informed by NHS Scotland that some confusion may have arisen in a few cases where people have called the National Helpline to see if they have an appointment scheduled ahead of receiving a formal invite by letter. On occasion there can be a slight lag between an appointment being scheduled and an appointment letter arriving on someone’s doorstep.”
Ms Freeman advised those aged 60 to 64 who have not yet received a letter not to call the vaccination helpline.
"Personally I would wait, maybe into the early part of next week, and if you haven't had your blue envelope then phone the helpline and see if they can help you in terms of knowing whether or not you've got a booking,” she said.
Ms Freeman announced that 1,844,636 in Scotland have so far been given a first dose of Covid vaccine, while almost 150,000 have received a second dose.
Some 46 per cent of 60 to 64 year olds have been given a first dose so far.
A Scottish Government spokesperson said: “NHS Lothian have informed us of a delay in the delivery of some vaccine appointment letters, however the Cabinet Secretary had not been advised of this when she gave the COVID media briefing today.
“As NHS Lothian have made clear, letters about appointments for next week are expected to delivered tomorrow.
"If the appointment date is not suitable it can of course be rescheduled by visiting NHS Inform or by calling the national helpline on 0800 030 8013. Anyone over 60 who has not received an appointment letter by Monday should ring the helpline.”