Boo1's comments are on target. But something about the proposed use of railroad rail suggests the wall is not very tall or significant. Or that the local government badly underestimates their problem and you may get stuck with the repair cost when your solution (using their materials) doesn't work.
First, a few questions. Where is your site? How long are the rails? What is the approximate section modulus? What are the approximate cross-section dimensions?
The importance of the wall is driven by several factors, such as wall height, what it retains, and the slope angle. If it is short (say, 1.5 meters or less), flat (less than 25 degrees from horizontal), nothing of significance is behind it and excessive wall movement will not impede roadway traffic, then the consequences of failure may be small. A "soldier pile and lagging" design (see Andy Machon's comments) in a moderately strong clay soil should work fine provided the rail penetrates the ground at least twice as far as its exposed height and fully penetrates the slope failure enough to reinforce the slope and prevent further movement. (That's a big "provided".) I wouldn't exceed a rail spacing of 2 meters; and a rail spacing of less than 1 meter may be required depending on ground conditions, backfill material and the problem's geometry.
If the wall height exceeds 1.5 meters, will be located where the public will walk or drive along it (and notice it is leaning), or will retain soil adjacent to a building or public space that cannot tolerate horizontal/vertical movements that could exceed 80 mm (3 inches), or potentially cause damage to a third party (like an adjacent property owner), pass this assignment to an experienced engineer.
A note of caution: you did not indicate the total height of the slope that this wall will be repairing. (This may be - and probably is - different that the wall's height.) Slope failures generate very large forces. If the total slope height exceeds about 2 meters, or is steeper than 3:1 (H:V), then used rail may not be appropriate.
Keep in mind that the existing failure may become YOUR failure if you aren't careful.