The South Korean striker has yet to score a league goal for the Premier League outfit since joining the team from Pohang Steelers one year ago.
Watford were keen on the 28 year-old before he made his move to Teesside but face stiff competition from Charlton.
"Already, Charlton have made enquiries about Lee to Middlesbrough," a source close to the player told Korean site Sports World.
"If he gets more playing time on the pitch then Lee doesn't mind if it's in the championship or Europe. He would prefer to do that than return to Korea."
The transfer fee could be a stumbling block. Although Middlesbrough picked up Lee on a free transfer, the Teesside team are believed to be looking for around £1.2 million.