In this paper, a construction of de Bruijn sequences using maximum length linear sequences is considered. The construction is based on the well-known cross-join (CJ) method: Maximum length linear sequences are used to produce de Bruijn sequences by the CJ process. Properties of the CJ pairs in the maximum length linear sequences are investigated. It is conjectured that the number of CJ pairs in a maximum length linear sequence is given by (2 +1)/3-2 , where n≥2 is the length of the linear feedback shift register associated with the sequence. The CJ pairs for some special cases are obtained. An algorithm for finding CJ pairs is described and a method of implementation is discussed briefly.