This work is devoted to the physical and mechanical properties of porous alloys based on TiNi alloyed with different amounts of Cu additive. We show that by doping a porous TiNi alloy with copper instead of nickel, it is possible to obtain characteristics acceptable for use in implantology and superior to those of known porous TiNi alloys. Cu addition in the range from 1 to 10 at.% is shown to optimize the properties of tested alloys. There is a decrease in the minimal martensitic transformation stress from 37 to 17 MPa when compared to initial unalloyed TiNi. Alloys with 3 and 6 at.% of Cu are found to be optimal for use in medical practice. Along with a wide temperature range of reversible deformations that cover the range of operating temperatures (273–313 K), such alloys demonstrate their martensitic transformation stress values below 28 MPs. This permits to model implantable structures of complex configuration from such materials under a certain temperature regime.