Good question! Someone more knowledgeable will need to give you the definitive answer, but I think the answer is "no." There may have been plans, but those two stretches of river fell very early in 1862. As I recall, both the CSS Arkansas and her sister ship, the CSS Tennessee were under construction at Memphis on the Mississippi River and they had to be moved (the Tennessee was destroyed in place and the Arkansas was moved further downriver to complete construction) after the fall of Island #10. That is just my recollection. Hope it helps until some real experts come along...
To the best of my knowledge, the answer is no -- with one exception.
Grant seized Paducah, KY when the Confederates under Polk and Pillow invaded Kentucky, seizing Columbus, on September 3, 1861. That gave the Union control of the mouth of the Tennessee River, so none of the Confederate gunboats on the Mississippi could ever get into the Tennessee (and thus not into the Cumberland). Nashville fell shortly after forts Henry & Donelson did in February and I have never seen any mention of a gunboat built there or in service on the Cumberland or the Tennessee; outside of Nashville, I doubt there was any place to build one.
The exception comes along in 1864. During the Johnsonville Raid, Nathan Bedford Forrest used his artillery to capture four Union vessels on the Tennessee River: the supply steamers Mazeppa, Anna, and Venus, as well as the gunboat Undine. Forrest repaired the Undine and the Venus, then put some of his troopers aboard. By using his artillery in combination with the Undine, he was able to close in on Johnsonville, fighting off six gunboats the Union sent after him. Forrest then used his artillery to destroy the Union depot (docks and warehouses) at Johnsonville -- shattered whiskey barrels in the warehouses unleashed a flaming river that rolled downhill, setting fire to everything it touched. Three Union gunboats plus 28 steamers and barges were destroyed or damaged, then set afire by the Union commander to prevent their capture. Venus was recaptured by the Union, but Forrest burned Undine to avoid her recapture as he retreated.
Undine was a "tinclad" with 8 x 24-pounder brass howitzers, taken into Union service in March 1864, captured by the Confederates October 30, destroyed November 4th.