Clarke received the £2m bonus on top of the £900,000 Open winner's cheque, which he won after securing his first golf major title at the age of 42.
The terms of the sponsorship between Clarke and Dunlop owner Mike Ashley was a no-win, no-pay basis.
The deal was struck six years ago between Ashley and International Sports Management, which also represents golfers such as Lee Westwood.
Northern Ireland's Clarke wore Dunlop clothing for the duration of the tournament.
Dunlop will hand over the cheque at a ceremony today (20 July) at the Lillywhites store in London.
Clarke, playing in his 20th Open Championship, was the first UK winner of the Open since Paul Lawrie in 1999.
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