Saxo-Tinkoff finished the 2013 Tour de France at the top of the team classification last weekend, but Riis said that finding common ground to renew the sponsorship deal with Tinkoff had proved impossible.
"Most importantly, we disagree on how the team should be run," Riis said in a statement, adding that his team would seek a new co-sponsor alongside Danish Saxo Bank for next season.
In a posting on Facebook, Tinkov said the split was because of poor results this year and a desire by his credit card company to go it alone.
"From January 2014, we will start to build our own cycling team Tinkoff Credit Systems. This will be (a) big project with 5 years commitment," said Tinkov.
"I want to have some control of the team, I don't want to just be the sponsor," Tinkov told Reuters, adding he could seek to construct a team from scratch or try to take over one.
Tinkov said teams were reliant on money from sponsors because very little cash from TV rights or race revenues was passed on to them.
Saxo-Tinkoff rider Alberto Contador had been one of the favourites to win the Tour but finished fourth behind Briton Chris Froome. Tinkov's personal Twitter account had carried remarks criticising the Spanish rider during the race.
Tinkov described the team's results this year as a disaster, saying it had made little impact outside of the Tour where Contador had fallen short of expectations.
The Russian is the founder of the Tinkoff credit card supplier which had a team that competed in the Giro d'Italia in the 2007 and 2008 seasons.
Reports before the Tour last month said that Tinkoff wanted to take over as sole sponsor of the Danish team after becoming a backer the previous year.
Cycling is seen as offering sponsors excellent exposure at a reasonable price if they are prepared to trust that the sport is putting its doping days behind it.
"The sport has probably never been as popular as now despite doping cases," Riis told Reuters.
"We are an international team so we are looking for a sponsor who can use that," he added.
(Additional reporting by Ole Mikkelsen in Copenhagen, editing by Justin Palmer)
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