Head coach Peter Moores, who recently returned to the England fold for a second spell, will name his first Test squad for the Sri Lanka contest in June.
"Look, the way I see it, it's pretty black and white," Sky Sports News quotes Gillespie as saying. "Jonny is the current Test keeper - simple. Explain to me what anyone else has done to allow them to not pick him? That's where I stand.
"All things being equal, Jonny should be afforded that opportunity. Jonny could be a very dynamic player coming in at number seven for England - that could be a role for him."
Matt Prior, who is also vying for that Test spot, lost his regular position as wicketkeeper during England's disastrous Ashes campaign in Australia last year.