Seasonal fluctuations are  often  found  in many  time  series.   In addition, non-linearity  and  the  relationship  with  other   time series   are  prominent behaviors  of  several,  of  such   series. In this   paper,    we  consider   the modeling  of multiplicative seasonal threshold autoregressive processes with exogenous input (TSARX), which explicitly and simultaneously incorporate multiplicative seasonality and threshold nonlinearity. Seasonality is modeled to  be  stochastic and  regime  dependent.  The  proposed model  is  a  special case  of a  threshold autoregressive process with  exogenous input  (TARX). We  develop   a   procedure  based  on  Bayesian  methods   to   identify  the model,   estimate parameters,  validate  the  model  and  calculate  forecasts. In  the identification stage   of  the  model,   we  present a  statistical test   of regime  dependent multiplicative seasonality.  The proposed methodology is illustrated with a simulated example and applied  to economic empirical data.