The plastic-encased circular device has a silicon strap that clips onto the top tube of the bike or the handlebars.
Once the user has set their end goal in the companion app, BeeLine uses the phone's GPS to give directions to the rider.
"Many of us can relate to journeys when we start cycling, only to stop a few minutes later to check the map on our phone," said BeeLine co-founder Mark Jenner, who together with Tom Putnam hired Map to design the product.
The LCD interface features a slightly rounded arrow that points in the direction of the final destination, and also shows the distance to the cyclist's goal.
The user can then choose their own route through the city using the compass as a rough guide.
"We felt this stop-start style of journey, following strict directions, was taking the fun out of cycling and one of the main reasons we started getting around the city on our bikes: the freedom," said Jenner.
"We wanted to create a device, that guided us, but still put us in control of our journey and encouraged us to find new and interesting routes.
BeeLine is designed to be shockproof and rainproof, with a battery that lasts around four weeks. The device's adjustable strap and aluminium buckle allows it to be clipped to keys or a bag when not in use.
The £60,000 funding goal has been surpassed, with 25 days of the Kickstarter campaign still to go at the time of writing.
Fellow London startup Onomo has also recently launched a campaign for a similarly minimal compass attachment, which would use flashing lights to provide directions to cyclists.
Map's previous projects include a kit of "building blocks" aimed at simplifying the creation of internet-connected products, a collection of wall storage products for the bathroom and a build-your-own computer kit.