In this paper, we introduce the concept of weakly tangential mappings. This concept is a generalization of the property (E.A.) introduced by Aamri and El Moutawakil in a paper published in 2002. We study the well-posedness of the common fixed point problem for two weakly tangential and occasionally weakly compatible self-mappings of a metric space $(X, d)$ which satisfy a variant of a contractive condition considered by Ćirić in a paper published in 2005. The results established here provide natural extensions and continuations to some results obtained by Ćirić, Sharma and Yuel and Guay and Singh.