The soft-top, four-seat SUV has a fully retractable roof system with a roll-over protection system.
Land Rover is working on a technical solution that would make it possible to open and close the soft top in about 12 seconds.
The convertible top would add minimal weight and have little effect on the vehicle's torsional rigidity, Land Rover said.
The concept keeps the production Evoque's rising waist line and the car's distinctive shape has been retained.
The new convertible body style has been achieved with minimal changes to weight and torsional rigidity.
John Edwards, Land Rover 's global brand director, said the convertible "explores the potential to further extend the nameplate's appeal by giving existing customers more choice, [while] at the same time reaching out to a completely new customer base."
The show car is based on the Evoque's Dynamique Plus version and is powered by a four-cylinder, 240-hp 2.0-liter gasoline engine.
Land Rover Design Director Gerry McGovern and his team developed the concept. McGovern said there will only be minor changes if the car makes it to series production.
Land Rover will decide before the end of the year whether to build the concept and an announcement may come at the Paris auto show.
If confirmed, production of the convertible will start in early 2013 in Land Rover's Halewood, England.
The Evoque went on sale in the U.S. market in October as a five- and three-door crossover and has been in hot demand -- there are waiting times as long as 60 days in some areas of the country.
Land Rover sold 2,244 Evoques in the fourth quarter.
The brand's total U.S. sales for 2011 sales were 38,099, up 20 percent from the prior year. In January, Land Rover sales were up 41 percent, and 680 Evoques were sold.
If produced, the Evoque convertible would compete with the Nissan Murano CrossCabriolet.
The convertible would likely be priced higher than the current models, the five-door has a starting price of $43,995 and the three-door is $44,995. Both prices include destination.
In a bid to win more women and younger customers to the brand, Land Rover launched production of the Evoque in July 2011.
It is the lightest and most fuel-efficient model in the Range Rover lineup.