I assume the wall is a brick veneer cavity type; if it is a solid brick wall the issues would be quite different.

It depends on the amount of bulging but it is something to be very careful about because the solution may be to rebuild the wall. The interior bulging is strange if it is a cavity wall. It definitely needs input from a professional.

If the house is brick veneer (one layer of brick over a covering a wood frame) it is not nearly as much of an issue as if the house is masonry (typically two layers of brick thick.

Are the interior walls bulging into the room or out the same way as the exterior bricks?

That would mark a separation of the layers in a masonry wall and depending on the size and relation to openings could be a significant structural issue.

If the wall is two wythes of brick it is possible to install bolts through the wall and pump grout into the cavity between the wythes to stabililze it. However, I have never seen this done for a full story, only for a foundation wall from grade to the sill beam.

This would have to be a pretty special house to justify the risk of an expensive repair.

Hire an engineer with experience in forensic structural investigation.

If both walls are insync with the bulging, it's possible that the wood structure the bricks are tied to is compromised. Insect or water damage could be the reason.
You would need an onsite inspection to determine the walls condition. They have cameras you can put into the wall through a 1/2" hole to check out the space.
Does the interior wall, "give" when you put pressure on it?
I had a situation in a customer's house once where the wood was totally eaten away from 4 feet above the floor all the way down to the foundation and 14 feet wide. You could actually create waves on the wall by pushing on it.
Ron

In Bardstown, Kentucky, there is a home owned by a relative of Stephen Foster, where he wrote "My Old Kentucky Home".
This home has large metal Stars on four sides of the home. Being from Texas, I asked about the stars.

Turns out there were cables going through the house which turminated through the stars. The cables were threaded for some distance on both ends. The cables and stars were used to pull together buldging brick walls.

Many older brick homes built in the early 20th century and earlier were actually three courses of brick. The outer and inner walls were vertical and an inner course was horizontal. You need to consider how the house is constructed to determine a fix.