This is going to depend on the course you do, what gear you stay in while cycling, how severe your bursitis is, etc. It's impossible to say for sure, particularly if your bursitis was caused by injury.
All that said, my basic suggestion would be "proceed with caution". It will be important to be as light on yourself as possible. The cycling motion is going to involve your hip almost as much as the running motion does, so you'll need to be careful not to overwork the injury. Under normal circumstances I'd very much encourage it, as proper exercise helps the healing process a great deal, but if you are already training on running it's important to consider the other extreme. Rest is needed for all training to take effect. I don't wish to warn you off; if done properly, it will likely help your training regimen a good amount, help keep your injury mobile and healing, and, most importantly, be fun. But nothing is less fun than hurting yourself.
If you have access to a pool, I think a safer way to cross-train would be to swim laps or do water aerobics. This will give you the loosening, aerobic workout you're looking for without absolutely requiring your hips to be in the picture. But if you don't, cycling should be fine as long as you keep in mind that it's a rest day and muscle exhaustion is not the goal.