The call for Trump to send a deal to the Senate for approval came after Trump and North Korean leader Kim Jong Un signed an agreement that focused largely on promoting denuclearization, but little on the issue of human rights that Trump stressed earlier in the year.
"If the president reaches a significant agreement with North Korea, I hope it takes the form of a treaty," Senate Republican leader Mitch McConnell of Kentucky said, according to Politico. "Which route that the administration takes will be up to them but I do believe it will need to come to Congress in some form."
Sending an agreement as a treaty would require bipartisan support from over two-thirds of the chamber.
Senate Republican Whip John Cornyn noted Trump should take a lesson from President Barack Obama's decision to sign the Iran nuclear deal -- which Trump withdrew from last month -- as legislation rather than a treaty.
"If you try to do things unilaterally or just with the executive branch, that they don't last long," Conryn said. "I'm not hung up on the form, but I do think it's important that Congress give it the 'Good Housekeeping' seal of approval, because I think that would ensure longevity."
Sen. Mark Warner, D-Va., said it was to early to speculate on a congressional vote on a North Korea deal, but also made mention of the Iran nuclear deal.
"I would think that this president who was so critical of the Iran deal would want Congress' approval if and when there's ever something to vote on," Warner said.