Hours earlier, Thune was re-elected chairman of the Senate Republican Conference, making him third in line in the party's leadership in the chamber.
Only a month ago, Thune said Trump should step aside for Mike Pence at the top of the ticket.
"That's behind us," Thune said on "Squawk Alley." "It was a painful and difficult campaign for a lot of us, but he's now our president-elect and we're looking forward to working with him."
As conference chairman, Thune is responsible for creating and implementing the communications strategy for the Republican Conference. Also on Wednesday, Sen. Mitch McConnell of Kentucky was re-elected Senate Republican leader, and Sen. John Cornyn of Texas was re-elected as Senate Republican whip.
When asked about Trump's transition process, Thune said the president-elect has made some good appointments.
He also said Congress will work to repeal the Affordable Care Act quickly, but he expects some features to be retained. He expressed support for the ban on insurance companies denying people coverage for pre-existing conditions, something Trump also recently suggested would remain in place.
On immigration, Thune said he doesn't think there is support for a broad deportation process.
"Border security will be a high priority. I think the wall will be a component of that," he said. "Whether that wall is a physical wall or a virtual or technological wall remains to be seen, but I do think it is an element of the discussion about border security that's important."