Here is the letter from Grant to Schofield confirming that Schofield did not send any messages to Grant during the Nashville situation backstabbing Thomas. In fact, Grant writes that he remembers a conversation with Schofield after Nashville, and that Schofield defended Thomas and the delay to attack:
New Tork, Aug 1 188\ Gen. J. M. Schofield.
Dear General:— Your letter of the 12th of July has just been handed me by Col. Wherry of your staff. I have read it carefully, together with the article from the "Toledo Democrat." The lapse of time since the event spoken of in that article is so great that I feel some hesitation in answering your letter, and the article from the Democrat, as I might do if I had access to the archives at Washington; but writing from memory I think I can say with great positiveness there was never any despatch from you to me, or from you to anyone in Washington, disparaging Gen Thomas' movements at Nashville.
On the contrary my recollection is that when I met you on your way to Wilmington, North Carolina, subsequent to the battle of Nashville, you explained the situation at Nashville prior to Gen Thomas' movement against Hood, with a view of removing the feeling that I had that Thomas had been slow. I was very impatient at that time with what I thought was tardiness on the part of Gen Thomas, and was very much afraid that while he was lying there at Nashville and not moving his army, that Hood might cross the Tennessee River either above or below the City of Nashville, and get between him and the Ohio River, and make a retrograde movement of our army at Nashville a necessity, and very much embarrass and delay future operations of the armies. Laboring under this feeling and impression I was telegraphing Gen Thomas daily, and almost hourly, urging him to move out and attack Hood, and finally became so impatient that I contemplated his removal and the substitution of another officer in his place; but this feeling on my part was not added to by any despatches from any person from the scene of action except those from Gen Thomas himself I have certainly no recollection of receiving any despatches from Nashville during the time spoken of in the Article in the Democrat from any person but Gen Thomas himself I feel very sure that if any despatches had been received from you, I should now recollect it, and I am free to say that it would have created a prejudice to your disadvantage if I had received such despatches.
This much you are at liberty to use in any way you may deem proper. The other reflections which the author of the article here alluded to against you I of course am not called upon to say anything in regard to. The fact is your subsequent promotions are proof positive that I entertained none of the views set forth to your disadvantage in this article.
Very Truly Yours, U. S. Grant