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The notion of isometric and unitary asymptotes was introduced for powerbounded operators in 1989 and was generalized in 2016--2019 by K\'erchy. Inparticular, it was shown that there exist operators without unitary asymptote.In this paper operators are constructed which are quasisimilar to isometriesand do not have isometric asymptotes. Also a contraction is constructed whichis quasisimilar to the unilateral shift of infinite multiplicity and whoseisometric asymptote contains a (non-zero) unitary summand.

On isometric asymptotes of operators quasisimilar to isometries