The Greek texts behind the King James version1 and the ESV2 agree here and actually show the present passive:
καὶ τῇ τρίτῃ τελειοῦμαι
Lit. And the third day I am perfected
Only a few English versions show this, including Young's Literal Translation and the Orthodox New Testament.
τελειοῦμαι (teleioumai) is the first person indicative passive of the verb τελειόω (teleioō), which is listed variously as "finish", "complete", "perfect", "accomplish", and "fulfill" in the various lexicons.3. It is exactly the same word that is used in John 19:28-30:
Μετὰ τοῦτο εἰδὼς ὁ Ἰησοῦς ὅτι ἤδη πάντα τετέλεσται, ἵνα
τελειωθῇ ἡ γραφή, λέγει· Διψῶ. σκεῦος ἔκειτο ὄξους μεστόν· σπόγγον οὖν μεστὸν τοῦ ὄξους ὑσσώπῳ περιθέντες προσήνεγκαν αὐτοῦ τῷ στόματι.
ὅτε οὖν ἔλαβεν τὸ ὄξος [ὁ] Ἰησοῦς εἶπεν· Τετέλεσται, καὶ κλίνας
τὴν κεφαλὴν παρέδωκεν τὸ πνεῦμα.
After this, Jesus, knowing that all was now finished, said (to fulfill the Scripture), “I thirst.” A jar full of sour wine stood there, so they put a sponge full of the sour wine on a hyssop branch
and held it to his mouth. When Jesus had received the sour wine, he
said, “It is finished,” and he bowed his head and gave up his
Cyril of Alexandria explains here:
I am being perfected means that of His own will He would endure the Passion upon the Cross, for the salvation of the world. He knew both
how and when He would endure death in the flesh ... Of His own will He
consented to suffer, as being well assured that by the death of His
flesh He would abolish death, and return again to life. For he arose
from the dead, having raised up with Him the whole nature of man and
having fashioned it anew unto the life incorruptible (Sermon 100 on
(This is not related to your main question, it might be noted that despite what is in the vast majority of translations, the Greek text does not actually say "Go tell that fox", but rather "Go tell this fox" - πορευθέντες εἴπατε τῇ ἀλώπεκι ταύτῃ. The understanding was that Jesus was referring to one of the Pharisees who gave the report and not to Herod - see e.g. Cyril of Alexandria, Commentary on Luke, Sermon 100. The only English translations that are faithful to the Greek that I have found are Young's Literal Translation and the Orthodox New Testament.)
1. Assuming Scrivener 1881 Textus Receptus
2. Nestle-Aland 28th edition
3. e.g. Barclay's Concise Greek-English Dictionary of the New Testament, Swanson's Dictionary Biblical Languages with Semantic Domains: Greek)