In the past, many image encryption schemes have been developed through the swapping operations at the different levels of granularity. These levels span bits, Deoxyribonucleic acid (DNA) molecules, pixels, blocks of pixels. In this study, a new scheme for the encryption of color images based on the DNA strands level scrambling (DNASLS) and chaotic system has been proposed. After a color image is input, it is decomposed into the red, green and blue components. After it, these components are merged to form a big single image. Intertwining logistic map (ILM) has been used for the random data which generates three streams of random numbers. These streams have been further manipulated in such a way that nine streams are spawned out of them. One stream out of these nine streams has been used for the generation of a key image. Two streams have been used to DNA-encode the big single image and the key image. Afterwards, the strands of DNA-encoded single image are swapped with each other. Four streams determine the addresses of two DNA encoded pixels for the selection. A yet another stream is being used to select a particular strand from the DNA strands. To create the diffusion effects, an XOR operation has been done between the DNA encoded image after the swapping of strands and the DNA encoded key image. Finally, the last and ninth stream has been used to decode the DNA-encoded pixels into the decimal form. Purely random numbers with no interdependence have been employed in the entire encryption process. The effects of plaintext sensitivity have been achieved through the incorporation of Secure Hash Algorithm-256 or SHA-256 hash codes. In the end, the experiment and the security analysis have been performed. The results of the validation metrics like information entropy(7.9973), average key sensitivity(99.61%) and mean absolute error(84.7158) demonstrate the security, defiance to the number of attacks and a potential for real world application of the proposed image cipher.