<p>This study demonstrates the analysis of heat transmission for the flow of two different kinds of viscoelastic nanoliquids towards a penetrable linear contracting sheet. Single-phase nanoliquid model has been adopted for the analysis. Convenient similarity transformations are employed on the governing equations to attain self-similar non-linear ordinary differential equations (ODEs). The closed-form analytical solutions are obtained for the velocity and temperature fields. Dual and triple solutions have been found, respectively, for the second-grade and Walter’s B nanoliquids, subject to some specific ranges of the flow quantities. Interestingly, the inclusion of nanoparticles extends the regions of respective dual and triple solutions for the second-grade and Walter’s B nanoliquids. The influences of numerous physical parameters on the flow fields and heat transfer are observed using graphical representations, and the consequences are deliberated physically as far as possible. It is revealed that the thickness of the momentum boundary layer enlarges with the viscoelastic parameter in the first branch of solutions, and it degrades in the second branch of solutions. Again, for both second-grade and Walter’s B nanoliquids, the temperature decreases with the enhancement of the Prandtl number and transpiration parameter, while the temperature rises with the augmentation of the volume fraction of the nanoparticles and power-law index with direct variation for both genres of nanoliquids. The dependences of the velocity and temperature gradients on the existing flow parameters are delineated in different tables. Finally, the temperature elevates when nanofragments are added in the base fluid by suppressing the unstable convective mode and enforcing the stabilized conductive mode of heat transfer.</p>