Bills have been introduced in each house of the Tennessee legislature to bar naturalized citizens from running for congress in a party primary. They are SB 1825 by Senator Brent Taylor (R-Memphis) and HB 2036 by Representative Johnny Garrett (R-Goodlettsville). Here is the text. The two bills are identical.
It is clear that this bill, if enacted into law, would violate the U.S. Supreme Court decision U.S. Term Limits v Thornton, which said that states cannot add to the qualifications for congress that are mentioned in the U.S. Constitution.