Interaction of single-chain nanoparticles (SCNPs) with lipid membranes is studied theoretically based on Monte Carlo simulations using the bond fluctuation model. SCNPs with a tunable elasticity represent an intermediate between rigid nanoparticles and flexible polymers. The degree of cross-linking between monomers of the precursor polymer is the key parameter that allows gradually tuning the properties of SCNPs. Both the alternation of lipid bilayers by SCNPs with different degrees of hydrophobicity and cross-linking and the effect of a lipid bilayer on the shape of the SCNPs are studied. Simulation results for SCNPs indicate an adsorption transition on the bilayer which is sharply peaked around a critical degree of hydrophobicity. At this specific value of hydrophobicity SCNPs can destabilize the structure of the bilayer inducing enhanced permeability for water and small solutes. With increasing degrees of cross-linking, however, membrane penetration and induced permeability get suppressed as the object's shape gets more conserved. Hydrophobic SCNPs can be fully embedded into the membrane, and the induced permeability increases with the degree of cross-linking. Soft hydrophobic SCNPs adapt their shape changing from spherical to disklike upon membrane insertion, which allows them to minimize the lipid bilayer disturbance, while at high degrees of cross-linking SCNPs keep a spherical shape and disrupt the bilayer at the rim.