The Yorkshire batsman hit 118 on day one of the game against Mumbai 'A' on Saturday but his place in the middle order for the first Test is not assured.
Bairstow said: "I can only score as many runs as I can, work as hard as I can in the nets and do everything in my power to put my case forward for that final selection.
"There's a massive amount of competition in that middle order, but there's only 11 people who can play.
"So you hope scoring a weight of runs in the warm-ups means you're pushing for a place in that side."
Eoin Morgan, who is also aiming to be named in England's middle order for the first Test in Mumbai, scored 76 on Saturday.