good question

Very hardly so, and for most (if not all) practical purposes the answer shall probably be "no" (that's just me, however...).

You could take the pain to precisely measure in detail the stray field distribution around the specimen and try to deduce the magnetisation distribution required in the material to produce it. This is an example of the so-called "inversion" problem. unfortuately the solution is not unique. Finding the adequate would need guidance by intelligent principles relating to your specific situation. (The problem is similar to the reconstruction of brain activity from magnetic measurements at the head's surface.)

Alternatively you would need a specially prepared rod with local sensors of magnetic field. You could in principle think of some NMR set-up such as used in NMR imaging in medecine (tomography). However, since your sample is a metal, the rf e/m fields might not sufficiently penetrate the material to make NMR tomography practical.

One way will be to put embedded loops, fully insulated, of different radii and measure the voltage induced. From this data you can deduce the magnetic field distribution in the rod. But you will have to design and produce the steel rod with insulated loops!

In which case, Nagendra, it is no more the same rod and shall most likely possess a different internal field distribution!

Kai, I see your point. But any measurement produces perturbation of the quantity being measured. In my suggestion above, it the the inguiniety of the designer of the so-called embedded search coil within the rod that would detrmine the success of the experiment.

As nan example, in plasmas magnetic fields of waves are measured by very sophisticated search coils. The induced currents in the search coils in the process of measuements do not significantly perturb the fields being measured. But plasma is a gas, the steel rod is solid, so embedding the search coils in a sample rod and implementing the measurement may not be a simple task. So honestly I do not know how practical this method could be!

Kai and Nagendra, thanks for your helping. The embedded coil method is a possible way to measure the magnetic field inside the steel rob, I've seen this method in a paper before. But, as Kai pointed out, the embedded coils are inside the steel rob, so they will change the local conductivity and permeability. Maybe if the coils small enough, this method can be a good approximation.

I guess you can try needle probe method. Instead of measuring total flux using the coil, it gives you a rather local measurement of your sample. I've tried needle probes to obtain magnetic flux distribution of a punched electrical sheet, you can check some of my publications:

In addition to the above comments

You can build your own electromagnetic(EM) coil and have a good measurement of the magnetic induction field intensity throughout the rod.

Here are few practical suggestions to build your own EM coil

1. Pick a magnet wire or 30AWG.

2. As the lady of the house for a sewing reel with inner diameter (ID) about half size of your rod OD.

 3. Winding the above wire into the reel by placing turns close together to make the first layers, follow that winding for the second layer and so on so forth.

4. Make the high of the above winding (aka # layers) about the OD of your rod.

5. Make sure you apply enough tension to the wires to have a very small but compact EM coil.

6. Apply ruban electric tape to the winding to increase compactness. The sensitivity of the coil depends on how compact these turns are.

7. Test continuity and do a measure ( use a voltmeter) resistance the coil. Note that coil must be short-circuited.    

8. Close the coil by connecting a resistor (1/4W) at the end of these terminals. 

9. Take scope prove and measure the voltage drops across that resistor and connect it to scope.

10. Apply either pulse current or constant current to the rod and you will have a transient displayed in the scope. 

Voila! You have there your EM prove.  

11. If you place your EM coil from the side and about the center of your rod, you should get zero reading because at that point the net flux is zero at that region. This is valid if the current that flows throughout the bar is homogeneous, though.

I hope it helps