"We are very confident that President Trump has the deal-making skills that are necessary in order to bring him home and to take a stronger stance with Iran and demand his release," Levinson's daughter, Stephanie Curry, told CNN's Alisyn Camerota on "New Day."
Ten years ago this week, Levinson disappeared in Iran. Friday is his 69th birthday.
Levinson's son, David, said he believes Trump will keep their father at the forefront of any future negotiations with Iran.
"President Trump has the negotiating skills and the assertiveness as well to make this a priority. If he makes it a priority in any discussion and he keeps this front and center of the conversation in any discussion with Iran, he can get the deal done," he said.
White House press secretary Sean Spicer said Thursday that the Trump administration is deeply committed to returning Levinson to his family.
"The Trump administration remains unwavering in our commitment to locate Mr. Levinson and bring him home," Spicer said. "The Levinson family has suffered far too long, and we will not rest until his case is resolved."
Trump repeatedly criticized former President Barack Obama for not securing the release of prisoners held in Iran as part of talks on the Iran nuclear deal. Levinson's children said they believe their father has been alive for the past 10 years.
"We believe in what he said on the campaign trail. He said he believes that Iran knows exactly where my father is -- and that's what we believe as well," David Levinson said.
"President Trump has pledged to bring him home and not to leave any American patriot behind. And our father has been repeatedly left behind before," Curry said. "Again, I say two previous administrations have failed and he can do something that nobody else has been able to do."
The Levinson family hopes to meet Trump soon to discuss his plan to return their father to the US.
"We would love to meet with him; we would love to sit down and have him outline the steps that he will take in order to bring our father home," Curry said. "It's been entirely too long."