This is often done when brand new aircraft are delivered to the customer, short haul aircraft in particular when crossing the Atlantic or Pacific can have “ferry tanks” fitted.

I think you can spot one slight issue with them, they take up an awful lot of room! So can they put them down under? Sure!

Although there are two notable downsides. First being you lose all or part of your cargo space, but this probably isn’t an issue for most private jets. The second being this will add A LOT of extra weight, slightly increasing the landing distance required ,but vastly increasing the takeoff distance. This may render some airports completely unusable when fully loaded.

The concept isn’t new, in WW2 they invented drop tanks, essentially a bomb shaped tank that could be jettisoned when engaging in a dog fight or when empty. This hugely increased operational range and the USA used them extensively in bomber escort missions.

Even modern jet fighters use them! They have pretty horrific fuel consumption, and need all the help they can get (the three bulbous “bombs”).

Had a couple questions so leaving the answers here.

Boeing/Airbus style ferry tanks are inflatable and they just have a lock seal filler just like on the underside of the wings on each bag.

The metal ones in the original answer, it is hard to see in the picture but they are actually very short (there are 4 tanks in that pic) and they turn them 90 degrees and they fit out the door! As there is no cabin fittings this isn’t an issue.