Word Reference says that "avvertire" can mean "to advise", but the meaning in the corresponding example provided by WR dictionary (La maestra ha avvertito mia madre dei miei pessimi voti) is different: "to inform". Is the former meaning really possible in that verb? If I understood Treccani and Collins well, there is no such meaning in "avvertire".
I would personally never use avvertire to mean advise. However, I don't think it is completely wrong either.
In the Grande Dizionario della Lingua Italiana one reads under avvertire
- Ammonire, avvisare; far consapevole, edotto; informare.
I think that this last meaning is where the link appears. If one looks in the Cambridge Dictionary, one sees that one of the meanings of advise is
[formal] to give someone official information about something.
They were advised of their rights.
It might be a far-fetched hypothesis, but I suspect that this is what is intended on Word Reference.