Mr Wallace told The Economist “it’s not going to happen” and there are “a lot of unresolved issues in Nato”.
He said the US wants Mr Stoltenberg to stay in the role.
He indicated that Mr Wallace was a “very qualified individual” but it “remains to be seen” who will get the job.
Before his trip to the White House, Mr Sunak said Mr Wallace is widely respected across the world.
“Ben does a fantastic job. He is a great Defence Secretary,” Mr Sunak said at the start of June.
“Ben is widely respected among his colleagues around the world.
“We’re one of the only countries that participates in every single Nato operation. We are widely perceived as a thought leader in Nato.”
Downing Street on Thursday refused to say whether Mr Sunak was disappointed to see Mr Wallace exit the running to take over at Nato.
A spokesman for the Prime Minister said the Defence Secretary’s focus would “continue to be on protecting our national security and supporting Ukraine”.
The No 10 official, responding to questions about the appointment of a new alliance general secretary, added: “We want a candidate who will keep Nato strong and ensure it can continue to deter and defend against existing and future threats.
“You will be aware of the appointment rules and how it is run. It requires a consensus across the alliance, not just one country.
“The Prime Minister has made clear the great job the Defence Secretary is doing and will continue to do.”