Conversion from neutron stars to strange stars as a possible mechanism of cosmological gamma-ray bursts (GRBs) has been discussed in previous works, although the existence of strange stars is still an open question. On the basis of this mechanism, we here outline an explanation of the connection between supernovae (SNe) and GRBs, which has got increasing evidence recently. An asymmetric but normal SN explosion leaves a massive ($geq1.8{rm M_odot}$) and rapidly rotating neutron star, which then converts to a strange star few days later, due to its rapid spindown. The accompanied fireball, which can be accelerated to ultra-relativistic velocity ($Gamma_0sim 100$) due to the very low baryon contamination of the strange star, flows out along the direction of the high-velocity SN jet and subsequently produces a GRB and the following low energy afterglows by interacting with the surrounding stellar wind. We will also expect a very luminous supernova like SN1998bw, if a large fraction of the conversion energy finally turns into the kinetic energy of the supernova ejecta.
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