The Englishman has been without a club since leaving Nottingham Forest at the end of his contract last summer.
Lingard, who spent 22 years at United before leaving Old Trafford in 2022, said joining Seoul was a new challenge and he was "ready to accept it".
"It's a new beginning for me, a new chapter in my career," he told reporters at a press conference after being greeted by fans at the airport.
"I just want to enjoy playing football again, putting a smile on people's faces and showing the Korean fans what I can do."
Lingard said he had "multiple offers" but they were "all verbal", and that he joined Seoul because they were committed to signing him.
Lingard played only 20 times in all competitions for Forest and featured just four times in 2023.
He said it took "two or three days" to negotiate the deal with Seoul.
The club did not disclose the length of his contract but reports said it was for two years.
"Everyone was happy with the deal and now I've finally signed I can concentrate on my football," said Lingard.
"I'm looking forward to getting back on the pitch."
Seoul, who finished seventh in the 12-team table last season, said they had signed "a player with the biggest name value in the 41-year history of the K-League".
Lingard scored Manchester United's winner in the 2016 FA Cup final but often struggled to hold down a regular place in the team.
He starred during a loan spell at West Ham in the second half of the 2020/21 season, only to find himself on the fringes again when he returned to United.
He revealed last year that he used alcohol to "try to take the pain away" as he battled mental demons caused by his stalling United career and his mother's bouts of depression.
He has scored six goals in 32 appearances for England and was in the 2018 World Cup squad.
Lingard will join Seoul's training camp in Japan ahead of the new K-League season, which kicks off on March 1st.
"I've come here to embrace the culture, learn new things and more about the K-League, to make it more global than it already is," he said.
"For personal goals, I have them written down back at home, but it's about the team, and I want to play for the team."
