True Seeing vs. "invisible" illusions
If you look at an "invisible" illusion you see it for what it really is: An "invisible" illusion, i.e. the image someone capable of seeing "invisible" things has visualized. At the same time the illusion would be revealed as such.
As the question also adresses the problem if and how such "invisible" illusions can be created, I would also like to discuss this.
It is not possible to cast invisibility on an illusion
The spell invisibility targets only creatures or objects not spells or spell effects.
It is possible to create "invisible" illusions.
A visual illusion can be "invisible". "Invisibility" in D&D game terms does not assume that something really cannot be seen. It basically says you are concealed from normal sight.
Invisibility: Invisibility makes a creature undetectable by vision,
even darkvision. DMG p. 295
On the other hand, the DMG defines rules for spotting the “presence” of an invisible creature or object and for even pinpointing it. The chapter concludes with the sentence:
DMG p. 296. Since some creatures can detect or even see invisible
creatures, it is helpful to be able to hide even when invisible.
Deducting from the last quote the first quote should be read “Invisibility make a creature undetectable by vision not qualified as detecting invisible creatures.” So we are not speaking about true invisibility but of being visible only to certain kinds of vision. Thus I put “invisible” in quotation marks.
The spell Silent Image says:
Effect: Visual figment that cannot extend beyond four 10-ft. cubes +
one 10-ft. cube/level (S)
This spell creates the visual illusion of an object, creature, or
force, as visualized by you. The illusion does not create sound,
smell, texture, or temperature. You can move the image within the
limits of the size of the effect. PHB p. 279
A figment is defined this way:
Figment: A figment spell creates a false sensation. Those who perceive
the figment perceive the same thing, not their own slightly different
versions of the figment. (It is not a personalized mental impression.)
Figments cannot make something seem to be something else. (…) you
cannot make a visual copy of something unless you know what it looks
like. PHB p.173
The part “those who perceive the figment” clearly states that a figment does not have to be perceived by all creatures. You can cast Ghost Sound when deaf creatures are around – they just do not hear it.
If the caster has cast See Invisibility or has the blindsight quality there is no reason why he could not visualize something "invisible". Thus he could use Silent Image to create an "invisible" illusion with no need for other spells.
The DM might even allow creating an “invisible” illusion if the caster is not currently able to see invisibility but has cast the spell in the past.