In this paper we examine stability of Lur’e-type systems arising as a feedback superpositions of infinite-dimensional linear blocks, described by integrodifferential Volterra equations, and periodic nonlinearities. Such systems have multiple equilibria, so traditional methods of stability investigation, defined for systems with single equilibrium are no good here. In the paper traditional Popov method of a priori integral indices is combined with two special techniques: Leonov’s nonlocal reduction method and the Bakaev-Guzh procedure. As a result new frequency–algebraic stability criteria are established, yielding tightened estimates of stability domains in the space of the system’s parameters.