If I remember correctly, in Game of Thrones Valyrian Steel is made with dragonfire. There are grown Dragons in Westeros, can the technique of forging Valyrian Steel be reproduced? Is this knowledge somewhere, for example the Citadel?
"the secrets of its making had been lost when the Doom came to old Valyria" (Tyrion IV, ASOS)
It's possible Sam has rediscovered some of those secrets from the Citadel - but there's been no mention of that yet.
Although no new Valyrian steel has been made, Tywin Lannister did have the Stark family sword reforged into two smaller swords. Tobho Mott, the smith who did the work, had Gendry as an apprentice. One fan theory is Gendry has returned in the show so that he can rediscover the secrets of Valyrian steel. Of course, right now he seems more interested in fighting.
These are just theories though - there's no-one we know now knows how to new Valyrian steel :)
Short answer : no If we know more or less how to reforge it without losing their capacity (and only the best blacksmiths can do it), the secret to create Valyrian steel was lost during the Doom (the cataclysmic event that cause the Valyrian Freehold to collapse). There is a magical process, other than just melting regular steel with dragonfire, otherwise the targaryans would have done it before.
Is this knowledge in Citadel is the question to which even grand-maesters can't answer. Could be it is in some old books, but it would require checking of everything, just as Sam suggested should be done only for White Walkers knowledge. But since that was Valyrian secret, could be that this knowledge is only in ruins of Valyria, not even in the Citadel.
As for Valyrian steel in general, answer is NO. It is spoken that dragonfire is needed for the process, but how precisely it was made is unknown. There is more to it than just applying dragonfire. Could be other ingredients are needed, could be different technique of forging and could include magic as well.
This might not prove anything, but Balerion the Dread melted swords to create Iron Throne. That didn't turn the steel into Valyrian steel. Those swords were already forged as steel, so if dragonfire is all that is needed then it should be applied during forging of weapons.... I just pointed that out just in case. Like I said, I doubt it is just using dragonfire on iron.