Wednesday's surgery had succeeded in stanching the bleeding, but a severe hip fracture and possible infection remained serious concerns, Lee Cook-jong said, according to the report.
The defector was rescued by South Koreans, who are treating him in a hospital.
Dozens of North Korean soldiers have fled to the South through the DMZ over the decades since the peninsula was divided, including two soldiers who crossed the frontier in June.
A spokesman for South Korea's Ministry of National Defense said on Thursday that no joint military drills involving all three countries had been planned.
Seoul's unification ministry said that the level of North Korea's denunciation softened compared with its immediate condemnation back in 2014 when Trump's predecessor Barack Obama visited Seoul.
The North's official media haven't reported the case as of Tuesday afternoon.
REUTERS US President Donald Trump says Kim Jong Un's North Korea is a"hell that no person deserves
While on average more than 1,000 North Koreans defect to South Korea every year, most travel via China and it is unusual for a North Korean to cross the land border dividing the two Koreas, which have been in a technical state of war since their 1950-53 conflict ended in a truce, not a peace treaty.
He defected from a North Korean guard post nearby and was being treated in hospital.
The drills, which will continue until Tuesday, allow South Korean and United States forces to practice air operations, air artillery strikes and aircraft carrier escorts. It is located inside the 4-kilometer-wide (2 1/2-mile-wide) Demilitarized Zone, which forms the de facto border between the Koreas since the Korean War.
However, there has been no exchange of gunfire between the two sides. A North Korean soldier defected there in 1998 and another in 2007 but neither of those events involved gunfire between the rivals, according to South Korea's military. But it is very rare for civilians to cross the closely guarded border with the South, which is fortified with minefields and barbed wire.
The senior diplomat added the USA was providing Seoul with sufficient information on the matter and that Trump's speech indicated Washington's acknowledgement of progress in dealing with North Korea issues.
The Joint Security Area was the site of some bloodshed during the Cold War but there hasn't been major violence there in recent years. The state-run Korean Central News Agency on Saturday criticized the US for increasing military tension in the Asia Pacific region, warning that the DPRK, as a responsible nuclear power protecting the gateway to Northeast Asia, would never stand by and watch as the USA attempted to bring about nuclear war.
United Kingdom to consider special status for United Kingdom woman detained in Iran
He will speak to the Foreign Secretary at the Foreign Office at 10am, in a bid to help secure his wife's release from Iran. The diplomatic protection issue was due to be explored at Johnson's meeting with Ratcliffe, the Foreign Office said.