Normal bases are used in coding theory, cryptography and other fields widely, and they play an important part in the study of finite fields. Let the prime p be the characteristic of the finite field Fq, n(≥2) be an integer and ξ a normal bases generator of Fqn over Fq. The existence of normal elements that satisfy some special conditions has always been one of the hot spots in the research about normal bases. Usually, Character sums are powerful tools for studying the existence of special elements in finite fields. We give a sufficient condition such that ξ and ξ+ξ-1 are normal bases generator of Fqn over Fq by character sums estimate, and obtain some lower bounds of q and n from the sufficient condition. When n=p=2, we give a exact counting formula of ξ satisfying the above-mentioned condition.