The class of 2K 2 -free graphs have been well studied in various contexts in the past. It is known that the class of {2K 2 , 2K 1 +K p }-free graphs and {2K 2 , (K 1 ∪K 2 )+K p }-free graphs admits a linear χ-binding function. In this paper, we study the classes of (P 3 ∪ P 2 )-free graphs which is a superclass of 2K 2 -free graphs. We show that {P 3 ∪ P 2 , 2K 1 + K p }-free graphs and {P 3 ∪P 2 , (K 1 ∪K 2 )+K p }-free graphs also admits linear χ-binding functions. In addition, we give tight chromatic bounds for {P 3 ∪ P 2 , HV N }-free graphs and {P 3 ∪ P 2 , diamond}-free graphs and it can be seen that the latter is an improvement of the existing bound given by A. P. Bharathi and S. A. Choudum [Colouring of (P 3 ∪ P 2 )-free graphs, Graphs and Combinatorics 34 (2018), 97-107].