The move will see both agencies unite under the glue London framework, which will continue to be led by Mark Cridge, the chief executive of glue.
De-construct is a creative digital agency that has offices in London and Amsterdam. Its clients include Alico, Chivas Regal and V Festival.
In December its managing director, Glenn Jones, left the agency to pursue other interests.
Anorak, another creative digital agency, has a client list that features Airmiles, Learndirect and Macmillan Cancer Support.
The agency was also responsible for creating the "Happiness Factory" campaign for Coca Cola.
Glue London has been part of the Aegis Group for around four and a half years, after the network paid £14.7m to acquire the agency in August 2005.