Yes you can! We have a range of options for varying sizes of dog starting from dog baskets that will fit either on a front rack or a rear rack for the smaller dogs, all the way up to front-loader cargo bikes that will carry even the biggest dogs.
I'll start by talking about the solutions we have for smaller dogs and then move up to more Mastiff-capable options.
The best option we have at the moment for small dogs is the Basil Buddy Basket. These baskets fit on a rear rack and they have a clip system, which fits into an adapter on your existing rack that clips on and off easily. They come with a tether, and a cushion inside the basket. They also have the option of adding a cage. If your dog is new to riding bikes, you might start with having the cage just to avoid them trying to jump out and as they get used to it, you might find that you can remove that cage and just use the tether. Over time, you might find that you can completely do without both the tether and cage. These are a really nice quality product that you can also use for other things like loading shopping bags when you don't have a dog in it. They can be quite multipurpose, since they have a handle on them, which pops up and lets you carry it like you would a picnic basket.
For small to medium sized dogs you might consider one of our Carry Freedom Trailers. These hook onto the back of most any bike or e-bike and come in two sizes. The addition of a plastic tub or wooden crate or dog-crate makes these a convenient way to carry your pup. Because they sit low to the ground dogs can be trained to jump in and out themselves without needing to lift them. We recommend adding a tether of some sort during the time your dog is getting used to it.
For medium to large sized dogs really the best solution is a front loader cargo bike. We have a number of customers who have bought front loaders so that they can not just do their personal errands but so that they can carry their dog with them. In my own family we have a Brittany which is a mid sized dog of about 20kg. He rides with my wife to work every day in the front loader. It takes a little bit of training getting dogs used to riding in the bike, we suggest things like putting a cushioned yoga mat on the floor of the box, so dog has something to dig their claws into and doesn't slide around on the wooden/plastic base. It's nice to be able to put the dog on the front and if your dog has been crate trained of course front loaders make the perfect crate.
We have had dogs as big as a Leonburger (60-70kg) in the box of our front-loaders.
Dogs will learn to jump in and out of front loaders on their own on command. You might start by tethering them into the box as they get used to it. Our dog certainly took a bit of getting used to it and gained confidence. Now when he jumps in he wags his tail and puts his paws up on the side of the box and lets the wind flap his ears about so he's pretty happy. It's a great way to get dogs around. You can head on down to the dog park or down to the beach without having to jump into the car. If you live in town it's great to be able to take your dog around with you in places where it's too dangerous to have your dog running alongside. Dogs can curl up and get comfortable in the box. Once they're really familiar with the space they're happy to sleep in there. So if you're taking your dog and bike camping, it's a great place to let them sleep. It's almost like a mobile kennel.
Bonus - if you wanna see our dog Charlie while he was getting used to riding in the front box, click here.