The Scottish course hasn't staged the event since 2009 - five years before the US President bought the resort - and Eric says that the major championship should return as a reward for his father's love of the sport.
He told BBC Sport: "It deserves it, and I truly think that my father's been a guy who also deserves it.
"There's no-one that's been better for the game of golf. He's been a cheerleader his entire life, because he loves it and he believes in it.
"In time that will be rewarded. We'll be given our shot and again it will be a great honour and a great testament to a great legacy when we do."
However, Eric doubts that Turnberry will get to host The Open again until Trump's second term in the White House ends in early 2009.
Eric - who is in charge of his father's real estate business - said: "I think chances are by the time an Open comes back around, his time in government will be gone and we'll be back to being a strictly commercial company."
He believes that the South Ayrshire course has been overlooked in terms of hosting The Open for political reasons.
Eric said: "I think the elephant in the room was, frankly, politics, for a very long time."
Open organisers The R+A stated earlier this month that "big logistical issues" around Turnberry's local infrastructure needed to be overcome before the championship could return but Eric has disputed those claims.
He said: "You could host the Open Championship on that course tomorrow. It's impeccable, and Turnberry is ready for it.
"I think any challenge can be worked through and solved with smart people."