The textual variants are ἰδὼν and εἰδώς. According to Tischendorf, ἰδών is the well attested variant.
ἰδών is a participle declined in the aorist tense, active voice, nominative case, masculine gender, and singular number. It is derived from the aorist tense verb εἶδον.
εἰδώς is a participle declined in the perfect tense, active voice, nominative case, masculine gender, and singular number. It is derived from the perfect tense verb οἶδα.
The present tense of both verbs is εἴδω/ἴδω. But, aorist versus perfect tense hardly changes the essential meaning of a verb/participle. So, why the difference among English translations? As Thayer (p. 172) notes in his lexicon,
So, the aorist εἶδον and perfect οἶδα, tenses derived from the obsolete present εἴδω/ἴδω, possess distinct meanings: "to see" and "to know," respectively.
Matt. 12:25 happens to use the same expression, εἰδὼς...τὰς ἐνθυμήσεις. However, like Matt. 9:4, there is a textual variant.
Here I believe the Codex Sinaitus favors εἰδώς (it has ϊδωϲ).
In any case, context seems to suggest εἰδώς since thoughts can't be perceived, but they can be known (well, read below).
There's a prophecy by the prophet Isaiah commonly accepted to be regarding the King Messiah.
In Isa. 11:1-3, it is written,
1 And a shoot shall come forth from the stem of Jesse, and a branch shall bear fruit from his roots. 2 And the spirit of Yahveh shall rest upon him, the spirit of wisdom and understanding, the spirit of counsel and might, the spirit of knowledge and fear of Yahveh, 3 and his smelling is with the fear of Yahveh, and he shall not judge after the sight of his eyes, nor shall he decide after the hearing of his ears.
א וְיָצָא חֹטֶר מִגֵּזַע יִשָׁי וְנֵצֶר מִשָּׁרָשָׁיו יִפְרֶה ב וְנָחָה עָלָיו רוּחַ יהוה רוּחַ חָכְמָה וּבִינָה רוּחַ עֵצָה וּגְבוּרָה רוּחַ דַּעַת וְיִרְאַת יהוה ג וַהֲרִיחוֹ בְּיִרְאַת יהוה וְלֹא לְמַרְאֵה עֵינָיו יִשְׁפּוֹט וְלֹא לְמִשְׁמַע אָזְנָיו יוֹכִיחַ
Translators vary diversely in their translation of וַהֲרִיחוֹ בְּיִרְאַת יהוה, but I believe i have translated it appropriately. The King Messiah smells (yes, this is what the verb means) with the fear of Yahveh. The previous clause states that the "spirit...of the fear of Yahveh" rests upon him. In other words, it is the spirit resting upon the King Messiah that enables him to smell with the fear of Yahveh.
Notice how the King Messiah functions as a judge (see also v. 4), but he is not limited to merely judging by the sight of his eyes or hearing of his ears. In other words, he has the ability to judge, well, supernaturally. How does he do this? With the spirit of the fear of Yahveh resting upon him. And, what does "smell" have to do with this?
In the Babylonian Talmud, Seder Nezikin, Tractate Sanhedrin, Folio 93b, it is written,
Raba said: He smells [a man] and judges, as it is written (Isa. 11:3-4), "and he shall not judge after the sight of his eyes, nor decide after the hearing of his ears, yet he shall judge the poor with righteousness, and reprove with equity for the meek of the earth."
רבא אמר דמורח ודאין דכתיב (ישעיהו יא) ולא למראה עיניו ישפוט (ישעיהו יא) ושפט בצדק דלים והוכיח במישור לענוי ארץ
Bar Koziba reigned two and a half years, and then he said to the Rabbis, "I am the Messiah." They said to him, "With the Messiah, it is written that he smells and judges. Let us see whether he smells and judges." As soon as they saw that he did not smell and judge, they killed him."
בר כוזיבא מלך תרתין שנין ופלגא אמר להו לרבנן אנא משיח אמרו ליה במשיח כתיב דמורח ודאין נחזי אנן אי מורח ודאין כיון דחזיוהו דלא מורח ודאין קטלוהו
So, with the Lord Jesus Christ, he didn't need to be a witness to one's deeds, or to hear testimony from witnesses. He already knew a man's thoughts and could judge with absolute certainty.