Despite not being a first team regular, he scored many important goals for the club under Graeme Souness. In 1987, Fleck was transferred for £580,000 to Norwich City. He scored 66 goals in 181 appearances in his first spell with the club, earning a call-up to the Scotland Squad winning 4 caps, including 2 caps at the 1990 World Cup. 'Flecky' attained hero status among Norwich supporters and was voted Norwich City player of the year in 1992 at the end of the final season of his first spell at Carrow Road.
He moved on to Chelsea for a fee of £2,100,000 just before the first season of the new Premier League got underway, at the time a record transfer outlay for Chelsea, and a record sale for Norwich. The move proved unsuccessful; Fleck scored just four goals in 48 appearances and was loaned out to Bolton Wanderers and Bristol City. Fleck re-joined Norwich for £650,000 in September 1995, but his performances were not as consistent as in the past, however, and he moved to Reading towards the end of the 1997-98 season, spending one season at the Berkshire club and scoring once in nine league games before injury ended his playing career.
After a spell with Gorleston where he won the Norfolk Senior Cup in 2001, he joined Diss Town who play in the Ridgeons League, as manager in the summer of 2002. He won the Norfolk Senior Cup again in 2003 and 2005. A poor run of form during 2006 saw him sacked on 18th October 2006.
Fleck returned to employment with Norwich City in May 2007, when he joined the club's scouting network.
Fleck's nephew John is currently a midfielder in the Rangers Under 19 side.