Anything is possible, and doors are one of those things that are usually straightforward but potentially surprising in cost and mess. What it means in the very simplest case (without knowing details) is pulling off trim, cutting a new hold nearby, pulling out the old door frame and moving it to the new hole, filling in the old hole with wood and wall material on both sides, re-trimming everything, painting everything, some work on the floor, moving steps, and you're done.

We all wish estimates were that easy. You may find wires and pipes to be moved, overhead structure that needs supporting, things in the way of where you want your door to swing, adequate pathways to configure, and so on.

You said "move", but it might be cheaper to just add a new door and keep the old; you wouldn't have to use it.

Do you have pics and/or a sketch showing both door positions, the walls, the steps, the way the ceiling joist run, and so on?

Yeah, a floorplan sketch would help.

It sounds like you want to eliminate the door we see in the pic (leading from garage directly into the hallway) and then add a door that leads from garage directly into the laundry room (the area we see on the left), thus using the laundry room as a pass through-correct?

Removing the door won't be so much a problem since it's just a matter of closing things up and dry walling. But will it leave an odd space in the garage once closed up since you're moving steps?

Is there room in the garage to move the steps to the new location?
How will a new door swing and function and will it get in the way of anything?

Do people other than family members use the current hallway door a lot now and, if so, do you want them going thru your laundry room? Is there wiring or plumbing in the area where you want to put the door?

It sounds like I have a very similar arrangement. In the back corner of our garage we have a landing area with two doors- one enters the utility room and one enters the hallway directly into the house. We use both all the time. I can't imagine being without either door. But, there are times I do think of closing off the "end of the hallway" door. Like you I have a lot of doorway openings. Because of that we have a little "play" in the end wall. We've put an angle brace in the corner of the garage to try to eliminate some of it.

One thing I like about the door leading directly from garage into hallway is that it provides a straight shot for moving furniture in and out of the house.

So, there are both the structural issues and the functional issues.

No mention of the need for a header over the relocated door?