We present a phenomenological theory describing the self-similar coarsening stage in the collapse of a long flexible homopolymer chain in a dilute solution upon a sudden quench. The approach is based on the "necklace" picture of the collapsing chain being composed of clusters separated by strands as demonstrated by computer simulations. The model represents a special class in the cluster growth problem where aggregation is driven by tension in the connecting strands. The mean cluster size in the free-draining limit is predicted to grow with time as s(t)similar to t(6/7(1+nu)) where nu is the exponent characterizing the initial conformation of the coil. The characteristic collapse time scales as tau(c) similar to N(1+nu)similar to N-1.6 in agreement with the Langevin dynamics simulations. Incorporation of hydrodynamic effects leads to tau(c) similar to N(1/3+nu)similar to N-0.93. For a realistic experimental situation the theory presented thus predicts a much faster collapse than suggested by self-consistent field calculations.