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 Perakim, 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 were not nevi'im but possessors of ruach ha-kodesh. On the other hand, both Rashi and Rabbenu Chananel in Megillah 14a list David as a navi (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 (which he actually quotes as a proof in his introduction to Avot): 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
See also the notes in the Machon ha-Maor edition of Perush ha-Mishnah for more sources in the Rambam on both sides.
It wasn't said clearly here but the difference is pretty negligible. We find on 3a in megillah that even Daniel is not called a Navi despite many clear ( and unclear) prophecies. Navi is "lashon dibbur" saying it over. Daniel was not commanded to do so. Mathematical formula ;Navi is roughly equal to prophet ( don't have sign on my phone). Similarly , was David only supposed to write his prophecy or also say it over. That he had visions of the future is absolutely clear. As said above, negligible difference.
So the Gemara in Sotah 48b identifies David as one of the 'early' prophets:
משמתו נביאים הראשונים מאן נביאים הראשונים אמר רב הונא זה דוד ושמואל ושלמה
From the time when the early prophets died. Who are the early prophets? Rav Huna says: This is referring to David, and Shmuel, and Sholomo
It brings in Vayikra Rabba 30:3 a clear example of him showing prophetic revelation - it says the following:
בְּשָׁעָה שֶׁהָיָה צוֹפֶה וּמַבִּיט שֶׁצַּדִּיקִים עֲתִידִין לַעֲמֹד מִמֶּנּוּ כְּגוֹן: אָסָא, יְהוֹשָׁפָט, חִזְקִיָּה וְיֹאשִׁיָּה, הָיָה קוֹרֵא עַצְמוֹ מֶלֶךְ
At the time that he would foresee and observe that righteous ones were to rise from him in the future - such as Asa, Yehoshafat, Hizkiyah and Yoshiyah - he would call himself a king....
See Rashi on Megillah 14a, who cites Sefer Halachos Gedolos in the name of Seder Olam, who counts David and Shlomo in the list of the 48 Neviim.
However the Gra in his emendations (ad loc.) takes David and Shlomo out of the count.
See also Zohar (II, 154a) that states that Moshe Rabbeinu was the only person in history to merit both prophecy and monarchy. This would theoretically exclude David and Shlomo.
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.