We consider the Swendsen-Wang dynamics for the two-dimensional Ising model at low temperature in the presence of a small negative magnetic field h and with plus boundary conditions at the boundary of an arbitrarily large square. We analyze in detail the tunneling from the metastable phase to the stable one. In particular, we obtain an upper bound on the tunneling time tp by explicitly constructing paths in the space of spin configurations that drive the system from the metastable phase to the stable one. In our analysis the transition takes place through the formation of droplets of the right phase inside the wrong one with side greater than a certain critical value lc. The values of the tunneling time and of lc coincide with those found for a single-spin-flip dynamics in finite volume by Jordao-Neves and Schonmann.