The musician was sued by Daniel Ramos after the photographer said West attacked him outside an LA airport.
Gloria Allred, the lawyer representing Ramos, says the case has now been resolved a week before the trial was due to start.
She has released a photo of the pair shaking hands and smiling.
The lawsuit said that West attacked Ramos in 2013 after the paparazzo asked the rapper why he wouldn't talk to photographers.
"Celebrities are not above the law, and they have no right to physically attack someone simply because they were asked a question," Allred wrote in a statement.
Shawn Holley, West's lawyer, has not made a comment about the settlement.
In a statement, Allred says the other details relating to the legal agreement will not be made public, but the apology was an important part of the deal.
West, 37, pleaded no contest last year to misdemeanour battery over the attack. Misdemeanour generally means a less serious crime, while battery refers to any kind of unlawful physical contact.
He remains on probation.
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The Grand Tour was inspired by an 18th Century tradition where the wealthy toured Europe's cultural sites.
Nottingham Contemporary, Chatsworth House, Derby Museum and The Harley Gallery are all taking part.
It is hoped a number of exhibitions, which run until September, will increase visitor numbers at the venues.
The two counties have been chosen due to their connections to the original grand tours, which helped shape the art collection at Chatsworth, in Derbyshire, and The Welbeck Estate, in Nottinghamshire.
Much of Chatsworth's collection was amassed by the Dukes of Devonshire. Items were either collected during the tour or were inspired by what had been seen.
Local figures like Lord Byron, whose ancestral home is Newstead Abbey, would have also taken part in the tour.
At Nottingham Contemporary, artist Pablo Bronstein will display hand picked objects from Chatsworth House's collection, including works by Rembrandt and Frans Hals.
A colossal Roman marble foot from the stately home will also be on display in Nottingham.
In Derby, The Wright Revealed exhibition focuses on two rare paintings by 18th Century painter Joseph Wright.
One of the paintings languished in a museum store room for more than 50 years.
The Grand Tour project has been co-funded by the National Lottery and the Arts Council to help create cultural destinations around the UK.
Kanye West has settled a lawsuit with a paparazzo with a handshake and an apology, the photographer's lawyer says.
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A £350,000 project is set to link four cultural venues across Nottinghamshire and Derbyshire in a bid to increase tourism and boost the local economy.