The Hugo Boss logo has appeared on the bodywork of the McLaren F1 cars and featured on drivers' and team members' uniforms for the past three decades.
The competition forms part of a wider campaign called 'Dress Me for the Finale', devised by Germany-based branding agency Liganova, which involves branding activity, retail promotions, and point-of sale-activity within stores.
The competition involves inviting participants across the world to design the F1 stars' race overalls on the Hugo Boss website.
It will run in each F1 hosting country during the races, with one design being manufactured for qualifying in that country.
The competition is currently running in Australia, following the start of this year's racing season in Melbourne last weekend (25 March). It will kick off in the UK on 7 April.
At the end of the F1 season in November, fans will vote for the best design, which will be worn by Button and Hamilton during the final race in Brazil.
Hugo Boss launched an exclusive range of clothing on 25 March to mark the 30-year deal, which is McLaren's longest-running sponsorship, with Button and Hamilton modelling the range.
Last week McLaren relaunched its team website, to coincide with the start of the season.