The Third Medium Contact (TMC) method [1] has recently regained significant attention due to its con- ceptual simplicity. Instead of explicitly defining contact boundaries, the space between two interacting bodies is filled with a so-called third medium. This auxiliary material is chosen to behave very softly in the absence of contact and to stiffen progressively as the two bodies approach each other. Current re- search focuses on stabilizing the third-medium elements, which undergo considerable distortion during deformation. The introduction of an additional material in the void region is conceptually similar to fictitious domain approaches such as the Finite Cell Method (FCM) [2], making TMC a promising candidate for integra- tion into the FCM framework. However, embedding TMC in the FCM poses challenges related to the immersed boundary representation and the presence of cut cells. In standard FCM, the fictitious domain is modelled with a very soft material, and the transition to the physical domain is handled solely through numerical integration, since as the extreme softness of the fictitious domain renders weak discontinuities negligible. In contrast, when applying the TMC approach, the material interface must be accurately resolved to ensure reliable contact predictions. Failing to represent this interface properly may compromise the quality of the contact solution. A potential candidate to tackle this issue is the local enrichment of cut cells [3], where additional shape functions featuring a weak discontinuity at the interface are introduced. These enriched functions allow strain jumps and thereby should improve the accuracy of the numerical solution. In this contribution, we discuss the challenges and possible solution strategies for incorporating the TMC approach into the FCM. REFERENCES [1] Wriggers, P., Schröder, J. and Schwarz, A., A finite element method for contact using a third medium, Comput. Mech., 52, 837–847 (2013), DOI: 10.1007/s00466-013-0848-5. [2] Parvizian, J., Düster, A. and Rank, E. Finite cell method – h- and p-extension for em- bedded domain problems in solid mechanics. Comput. Mech. 41, 121–133 (2007), DOI: https://doi.org/10.1007/s00466-007-0173-y [3] Joulaian, M., Düster, A. Local enrichment of the finite cell method for problems with material interfaces. Comput. Mech. 52, 741–762 (2013), DOI: 10.1007/s00466-013-0853-8