Is there any suggestion in the scripture (even implicitly) that King David might have been also a Navi?
Whether or not David was a navi seems be an age-old discussion among Jewish authorities.
On the one hand, there seem to be many futuristic concepts throughout Tehillim, especially about the destruction of the Temple and the eventual redemption. Assuming that David was the author of at least most of the psalms would imply that he had at least some form of prophecy.
On the other hand, though, we don't ever find that God "spoke" to David as he did to other prophets. In fact, when God wanted to give David a direct message, it was sent either by Gad HaChozeh or Nassan HaNavi. This implies that David himself, at least at that time, was not a navi.
To find out what different people belive about David's prophetic status, a good place to look is the commentaries at the beginning of Sefer Tehillim. It is there in Ibn Ezra's introduction that you will find a list of sources in which the scipture "suggests" that David indeed was a navi. (The context there is actually sources that imply David as the author of all of Tehillim.) To give you the first couple:
- "כִּי כֵן מִצְוַת דָּוִיד אִישׁ הָאֱלֹהִים" [A description we find only by neviim.] (II Divrei Hayamim 8:14)
- "רוּחַ יְהוָה דִּבֶּר בִּי וּמִלָּתוֹ עַל לְשׁוֹנִי" (II Shmuel 23:2)
Whether David was a navi seems to be a machloket between the Rambam in his earlier writings and the Rambam in his later writings. In Shmonah prakim, the Rambam's introduction to Avot, ch. 7, he counts Shlomo and David as Nevi'im. However, in the Moreh Nevuchim, II:45, he says that they are not nevi'im but possessors of ruach ha-kodesh. However, both Rashi and Rabbenu Chananel in Megillah 14a count David (even if R. Chananel doesn't count Shlomo). The Rambam also adduces various proofs, e.g., in the Moreh, he counters an argument from שמואל ב', כ"ג, ג ואילו דברי דוד: אמר אלהי ישֹראל, לי דִבר צור ישֹראל, משמעותם שהוא הבטיח לו על-ידי נביא, או נתן או זולתו, כמו ויאמר ה' לה
Although David uses a navi to find out what Hashem wants(Natan) there is also some thought that David is a Navi in his own right. It appears that he uses Natan and others to find out what Hashem thinks about David's life, giving him parables to correct David's behavior while David speaks directly to Hashem in regard to war and the government.