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Is it possible to call an external linear solver from Abaqus and/or Ansys?
This solver (which is supplied by me) would get the sparse matrix A and the right hand side vector b as inputs, and would return the solution x (with A x = b) to the program. Ideally, Abaqus/Ansys would also inform the solver about the desired algorithm (CG, BiCGstab, GMREs, ...) and preconditioner.
Since Ansys 14.5 you can use ACT (Application Customization Toolkit) to connect external solvers and retrieve results.
As far as I know, there is no way to replace the ABAQUS solver with a user-written one.
However, you can use the *MATRIX OUTPUT option to write the stiffness (or mass or damping) matrices or load vectors to a text file. Then you can read these text files and call your solver.
Abaqus and ANSYS are "solvers" actually. Since they are closed source, and no modification is possible, so it would be better to shift your focus to open source programs e.g. Calculix for this purpose where you can modify the code to use your own solver, with very small effort.
Calculix uses ABAQUS's input file format, so you can use your input files (originally written for Abaqus), with Calculix.
Currently Calculix uses SPOOLES for solving sparse linear equations systems. You can easily replace that by your own solver.
