Born in Bovenkerk, Amstelveen, Bosman began his professional career at Ajax, making his Eredivisie debut on 20 November 1983 at the age of 18 in a 5–2 home win against Roda JC, scoring in the match.

Additionally, Bosman netted in double digits in both of his seasons with the Yellow Reds, notably scoring 18 in his first as the team won the national championship after a 41-year wait.

After one year back in his country with PSV Eindhoven, as successor to Wim Kieft who signed with a team in France – he served mainly as a target player for Romário, but felt uncomfortable in that role, and was also under threat of losing his starting position to youngster Twan Scheepers – Bosman returned to Belgium and joined Anderlecht, reuniting with former Mechelen teammates Graeme Rutjes and Bruno Versavel (and eventually de Mos) and being dubbed "The Giraffe".

Aged 31, Bosman returned to the Netherlands and signed for Twente, scoring 20 goals in his first season as the Enschede side finished third and qualified to the UEFA Cup, and being regularly used for the remainder of his spell.

Greece were extremely unhappy with the decision, accusing UEFA's West German chairman of preferring a Dutch team in the finals, bringing a larger crowd of supporters to the stadia; after the protests, the Hellenic Football Federation decided to move the last group match to a small stadium in Rhodes and field a sub-standard team – Netherlands also won that game (3–0).