We've traveled through Tanzania last year and we stayed at the Jimbiza Lodge in the Selous Game Reserve. It lies directly at the Rufiji river and it's just 10 minutes from the Mtemere gate (park entrance) away. You can also walk by foot into the next small village.
The staff is really friendly and helpful!
You can stay in safari tents, which have a view onto the river. There is just a row of bushes in the way. The tents cost around USD 235 per person (inclusive full board). Which isn't really cheap of course. But it's what you have to pay also everywhere else to get that standard.
But you can also use their campsite:
USD 15 per person with own camping gear
USD 25 per person using lodge camping gear
USD 10 per meal BBQ
USD 5 per person access to swimming pool
There are an array of activities which Jimbiza Lodge Selous offer to their clients: Game drives, Boat safaris, Birds watching, Walking safaris & Village cultural safaris.
As there are no fences around the Selous Game Reserve, you'll also be able to see wild elephants and other animals on the ground of the Jimbiza Lodge.
We did the boat safari which I can really recommend for USD 35 per person for 4hours. Here is the video to it: traveldudes.org/travel-videos/full-wildlife-selous-game-reserve-tanzania/1044