Madrid hosts the largest concentration of universities in Spain, with public universities being very prominent.

UNIVERSIDAD COMPLUTENSE DE MADRID:  the largest Spanish university, it is a comprehensive university that offers a large variety of degrees and has excellent international links and campus and the largest library in Spain, only after the National Library.
UNIVERSIDAD AUTONOMA DE MADRID:  Excellent connections with business and international partnerships, the Autónoma is a modern, large university and a reference in research. https://www.youtube.com/watch?v=vLcuECZK1No 
UNIVERSIDAD POLITECNICA DE MADRID:  within the top 100 worldwide in engineering and technology, the Politécnica is highly innovative and international, offering students excellent employment opportunities.
UNIVERSIDAD DE ALCALA DE HENARES: a world heritage university, Alcalá is ranked best in Spanish studies.  On top of this, it has modern facilities, is well connected internationally, known by the quality of its research and offers a varied portfolio of degrees
UNIVERSIDAD CARLOS III: a new university which year on year features in the top 50 universities.  With 40 per cent of its PG students being international, it offers many qualifications in English and has a fantastic Business School.
UNIVERSIDAD REY JUAN CARLOS: the youngest of the Madrid public universities, it has a variety of quality degrees and PG, the ambition, resources and modern facilities.

Together, the public universities in Madrid have agreements either for Erasmus or for other matters with nearly all UK universities.  As it would be really hard to re-negotiate agreements individually, they are launching,  together with the Regional Government in Madrid, this call for expressions of interest with a double objective:

a) to engage in bilateral partnerships with some or all of the above universities for exchange, collaboration and students/staff mobility programmes post Brexit.  These could include UG and PG programmes.
b) to form part of a working group that revisit current circumstances and opportunities/options in both countries to prepare a variety of possible agreements so that these can then be signed by those universities in a) above.