The move comes to “help get ahead of any significant rise in cases” due to the variant first identified in India, NHS Lanarkshire said.
Capacity at vaccine clinics has been ramped up to hit an target of 60,000 doses a week.
Currently, more than three-quarters of all adults in the UK have had their first Covid-19 jab. Almost half have had their second.
NHS Lanarkshire Covid vaccination lead, Dr Mark Russell, said people due their second dose at the end of June or beginning of July will see their appointments brought forward to mid-June.
He said: “To help get ahead of any significant rise in cases of the new Covid-19 variant, we are accelerating the rollout of our vaccination programme.
“From Monday, the number of people attending our vaccination centres will significantly increase.
“At the peak, we will be administering around 60,000 doses a week – this will be a combination of first doses for the younger age groups and second doses.
“To help continue on our road back to normality we need to make sure as many people as possible are vaccinated.
“It is also important to attend for both doses, for longer lasting protection.”
He asked people to only turn up with an appointment, cancel if they could not make it, come close to their appointment time to avoid queues, and to come on their own.
Dr Russell added: “We need to balance how we deal with the new variant of Covid-19 while continuing with the recovery of other NHS services.”