It appears that they are. You would be advised to have a gunsmith check the fit before taking it to the field or range.
The Model 37 is a pump gun, and the 51 is a semi-automatic. I doubt seriously that the barrels are interchangeable, since they are totally different models.
First off, the Ithaca is one of the finest pump shotguns ever made, and the ones made now are the best yet. (And no, I don't work for them, but I shoot them!) The first criteria is how old is you Ithaca Model 37. Ithacas with a serial number lower than 855,000 DO NOT have interchangeable barrels, so for those, the answer would be no. Ithaca M 37 shotguns with a serial number higher than would easily swap out both barrels and magazine tube. Make sure you have the right spring to go with the 8 shot tube and you should be set. Any more questions can be directed to the Ithaca website in the related link below.
Watch Gunbroker.com - they often have barrels listed. Also you might try Ithaca Gun USA, a company that has picked up the rights to manufacture the Model 37.
What is your question? Age? Value? History?
A Model 37 is a "pump", so a double barrel could not be a Model 37
1947
Made in 1949
Made in 1968
Assuming you are referring to a Model 37 pump, it was made in 1959
Bicentennial ithica 12 guage value
Most all the Ithaca Pump model 37 were Engraved with 3 ducks and on the other side Pointer Flushing out 2 pheasants. It a common engraving we all see on Ithaca but you have an 1937 model like I do ,I have one as well with a serial number 2493, believe me they are worth keeping because that was the 1st year Ithaca made the Model 37 they only produced 3,500 shotgun in 1937
Most all the Ithaca Pump model 37 were Engraved with 3 ducks and on the other side Pointer Flushing out 2 pheasants. It a common engraving we all see on Ithaca but you have an 1937 model like I do ,I have one as well with a serial number 2493, believe me they are worth keeping because that was the 1st year Ithaca made the Model 37 they only produced 3,500 shotgun in 1937