Yes you certainly can.
Pertinent text for adding spells to your spell book PHB pg. 94:
The spells that you add to your spellbook as you gain levels
reflect the arcane research you conduct on your own, as well
as intellectual breakthroughs you have had about the nature
of the multiverse.
Note that there are no restrictions on how you come about this knowledge. Most examples in this section cite finding written sources, however any research you conduct on your own could also qualify. Examples of research include practicing with other spellcasters, experimenting on your downtime with random components, studying books, talking to other casters, debates, etc.
Basically, it's up to you and the DM to come up with a way for your Wizard to add spells. The typical method is scrolls and books, but you aren't limited to just that. Since you are capable of adding spells through research, which is not limited to scrolls and books, then speaking with a Bard who has spells you do not, and having them show you how to do the spell, can indeed qualify as a method of adding new spells to your book.
Further supporting evidence can be found on PHB pg. 203 with reference to the schools of magic.
The Schools of Magic
Academies of magic group spells into eight categories called
schools of magic. Scholars, particularly wizards, apply these
categories to all spells, believing that all magic functions in
essentially the same way, whether it derives from rigorous
study or is bestowed by a deity.
The inference here is fairly clear. A wizard doesn't see a Bard spell functioning any differently from a Wizard spell. This is also clearly highlighted under the Wizard text oh PHB pg. 112, Scholars of the Arcane (emphasis mine):
Wizards live and die by their spells. Everything else
is secondary. They learn new spells as they experiment
and grow in experience. They can also learn them from
other wizards, from ancient tomes or inscriptions,
and from ancient creatures (such as the fey) that are
steeped in magic.
So clearly a written version of a spell is not required in order to scribe one. Nevermind the very obvious inference from reducibility, in that the first wizard would have had no written references to scribe their spells from. This means that spells can and are able to be scribed from sources other than written mediums.
Lastly, the wizard itself serves as an example of something able to scribe a spell without having a written scroll or book to reference it from. If a wizard loses their spellbook, they can create a new one without having a written reference:
Replacing the Book. You can copy a spell from your own
spellbook into another book—for example, if you want
to make a backup copy of your spellbook. This is just like
copying a new spell into your spellbook, but faster and easier,
since you understand your own notation and already know
how to cast the spell. You need spend only 1 hour and 10 gp
for each level of the copied spell.
If you lose your spellbook, you can use the same procedure
to transcribe the spells that you have prepared into a new
This last bit is important because you aren't using anything written yet you are scribing a spell anyways.
However you want to look at it, your Wizard does not REQUIRE a written form of a spell in order to scribe it. The wizard simply needs the spell to be on their spell list, they need some way to learn it, most commonly a written form, but it could be an instructor capable of casting a spell on their class list, or it could be simple experimentation on their own using a DC set by their DM to reflect the difficulty of the spell.
The only thing the rules state you have to do is:
- Spend 2 hours per level of spell (unless a class feature says otherwise); and
- Spend 50 gold per level of spell (unless a class feature says otherwise).
This is all covered under PHB pg. 114, Copying a Spell in the Book:
Copying a Spell into the Book. When you find a wizard spell
of 1st level or higher, you can add it to your spellbook if it is
of a level for which you have spell slots and if you can spare
the time to decipher and copy it.
Copying a spell into your spellbook involves reproducing
the basic form of the spell, then deciphering the unique
system of notation used by the wizard who wrote it. You
must practice the spell until you understand the sounds
or gestures required, then transcribe it into your spellbook
using your own notation.
For each level of the spell, the process takes 2 hours and
costs 50 gp. The cost represents material components you
expend as you experiment with the spell to master it, as well
as the fine inks you need to record it. Once you have spent
this time and money, you can prepare the spell just like your
None of this says anything about you having to have a written version of the spell to copy it at all. It says you have to decipher the unique notation of the wizard who wrote it? Great. What if a wizard didn't write it at all? Step done, no deciphering needed, next step. Practice hand gestures? No somatic component. Step done. Practice verbal? Good, verbal component only spell. Nice and easy. Practice away my friend.