Abercrombie House was opened by Lord Carrington in 1981 as part of an expansion of the Overseas Development Administration.
It later became the Department for International Development’s joint HQ before merging with the FCDO in 2020.
The department is due to redeploy 500 additional staff to Scotland by 2025.
No current site has been chosen for its new home, but an FCDO spokesperson said it would be a “modern building”.
They added: “The FCDO is growing its presence and number of staff in Scotland.
“HMRC will be moving into Abercrombie House, which is a building that is already part of the UK Government estate, providing better value for money to the taxpayer and which will continue to boost the economy of East Kilbride.
“HMRC is still at the early stages of plans for the building and hopes to confirm timelines in the spring.”
Council Leader Joe Fagan said: “I am frankly astounded by the announcement on the FCDO – both the decision itself and the way it has been made.
“These staff have been working, and in many cases living, in East Kilbride for decades and are important contributors to our local community and economy.
“The FCDO announcement has been made out of the blue, suggesting not just a failure of communication but also a lack of coherence in their decision-making process. The reasons given for the decision are also weak, to say the least.
“We call for urgent further discussions on this situation.”