"We were all delighted that Stewart signed a new contract last season. We believe he is happy and settled here," Lamb told the club's official website.
"Having said that, we know that Stewart has to achieve all that his talent allows, so the onus is on Middlesbrough to ensure his ambitions are fulfilled here.
"Stewart's got everything as a player and is playing possibly the best football of his career, both for Middlesbrough and for England.
"But I don't believe he has fulfilled his potential yet. His best years are ahead of him.
"In fact, we believe Stewart will become one of the best footballers not just in Britain but in Europe."