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 in the English Football League First Division. 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 helped them finished fourth in the league in 1989, when they also reached the FA Cup semi-final, though Fleck and his team-mates were unable to compete in the following season's UEFA Cup due to the ongoing ban on English clubs in European competitions that followed the Heysel disaster of 1985. Fleck helped the Canaries reach another FA Cup semi-final in 1992, where they surprisingly lost to Second Division underdogs Sunderland.
He moved on to Chelsea for a (then) club record fee of £2.1million just before the first season of the new Premier League got underway. It was also 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. He missed out on a place in Chelsea's squad for the 1994 FA Cup Final, which they lost 4-0 to Manchester United.
Despite his dismal goalscoring record at Chelsea, he is remembered fondly by the club's fans, who sang a song in his honour - We all live in a Robert Fleck world - based on the lyrics of Yellow Submarine by The Beatles.
Fleck re-joined Norwich (who had just been relegated from the Premier League) for £650,000 in September 1995 after a loan spell, but his performances were not as consistent as in his first spell at Carrow Road, as they failed to make a serious attempt to push for promotion in any of his three seasons back at the club. He moved to Reading towards the end of the 1997-98 season, spending one season at the Berkshire club, and in nine league games he scored once against Luton Town (in the first game at the Madejski Stadium), 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 18 October 2006.
Fleck returned to employment with Norwich City in May 2007, when he joined the club's scouting network.
Fleck is fourth in the list of Norwich City FC all-time goalscorers, behind Johnny Gavin, Terry Allcock and Iwan Roberts. In 2002, Norwich fans voted him into the club's Hall of Fame.
Fleck's nephew John plays for Rangers F.C..