Many researchers work with essential oils for several applications, such as their impregnation in polymeric films and for obtaining micro-/nanocapsules by using polymers as shell material. Polymer swelling/liquid dissolution is an extremely important property for controlled release of essential oils. Therefore, this work proposes a comparative study of swelling/dissolution interaction of polymer films immersed in different essential oils. Six polymers were used for this study: polycaprolactone (PCL) and poly(lactic acid) (PLA), gum arabic, sodium alginate, gelatin, and carboxymethyl cellulose. Polymer films were obtained by applying the casting method. In addition, PCL and PLA were molded by compression. Six different essential oils were used: Mentha piperita, Cymbopogon nardus, Mentha arvensis, Syzygium aromaticum, Lavandula hybrida, and Eucalyptus globulus. Each film was kept at room temperature immersed in sufficient quantity to cover the polymer film at intervals of 1, 3, 5, 7, 14, 28, 56, and 112 days. The same procedure was repeated at 4 °C and 35 °C at intervals of 1, 3, 5, and 7 days. PCL and PLA films were completely dissolved in Syzygium aromaticum essential oils for 24 h at the three temperatures tested. For all other polymers, weight gain occurred during the first 7 days and remained constant during the following days. The highest weight gains were 15.519% and 10.463% for polycaprolactone with Mentha piperita and gum arabic with Eucalyptus globulus, respectively.