The description of Truesight says:
A monster with truesight can, out to a specific range, see in normal and magical darkness, see invisible creatures and objects, automatically detect visual illusions and succeed on saving throws against them, and perceive the original form of a shapechanger or a creature that is transformed by magic. Furthermore, the monster can see into the Ethereal Plane within the same range.
The True Seeing spell description says:
This spell gives the willing creature you touch the ability to see things as they actually are. For the duration, the creature has truesight, notices secret doors hidden by magic, and can see into the Ethereal Plane, all out to a range of 120 feet.
The Detect Magic spell description says:
For the duration, you sense the presence of magic within 30 feet of you. If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.
Though I realize these are different things, I cannot help but notice the common trait of the awareness of magic within, or around objects or creatures. While Truesight and True Seeing seems to unravel the magic and is able to see beyond it, Detect Magic is only able to sense that something is magical until it uses an action. Given that all three have awareness about the magical world, would a creature that has truesight, or one that has True Seeing cast on it, also be able to tell if a weapon or object has a magical property or not?