The short answer is that there is no saying attributed to Jesus in the canonical gospels in which Jesus claims to be the King of Israel. The theory and practice of religion, however, is not an exact "science"; hence, the answer to this question, like all things in religion, depends on whom you ask. So, I will put the question to the author of the Gospel of Mark. All references to Jesus as king in Mark appear in chapter fifteen in connection with a hearing before Pilate and the crucifixion. At that hearing the first question Pilate asks Jesus is: "Are you the King of the Judeans?" (15:2, Ioudaiōn). Jesus responds evasively: "so you (sing.) say" (15:2). Pilate chides him for failing to give a direct answer to the question in view of the numerous charges brought against him (15:3-4). But Jesus remains silent (15:5). Pilate is astonished at his refusal to answer the question directly.
Pilate appears to attribute the information that Jesus claimed to be a king to the Judean people themselves (15:9, 12). The soldiers ironically mocked him as King of the Judeans (15:18) and the inscription of the charge against him read simply "The King of the Judeans" (15:2). That the kingship of Jesus in Mark is an idea that came from the Judean people is confirmed by Mark 15:36, when the chief priests and scribes mock him saying, "Let Christ, the King of Israel, come down from the cross" (15:32). By this mocking statement the religious leaders expanded the area of his kingship to include the whole of the traditional borders of Israel.
During the career of Jesus, Idumea, Judea, and Samaria constituted an Imperial province, governed by a Roman procurator (Pontius Pilate). The son of Herod the Great (Herod Antipas) governed at Rome's behest as tetrarch of Galilee and Peraea, and another of Herod's sons (Philip) governed at Rome's behest as tetrarch of the area east of the upper Jordan from Mt. Hermon to the Yarmuk River.1 The only actual king over the area was Caesar. Therefore, if Jesus were a claimant to the throne of Israel, he would have been in competition with Caesar Augustus for the status of king (as noted in John 19:12).
Mark reports no sayings in chapters 1-14 where Jesus overtly claims to be king and there are no sayings with a political edge to them. "King" Herod Antipas did not fear Jesus as a rival (6:14-29), and apparently not even the Judean people were thinking of him as a king (8:27-30). At the entry into Jerusalem the exulting crowd did not refer to him as a king (11:9-10). There are only two sayings attributed to Jesus in Mark that have political content. One is a factual statement about the power of rulers (10:42) and in the other Jesus supports the paying of taxes to the Roman government (12:13-17). Where then in Mark 1-14 does the idea that Jesus claimed to be King originate in order to account for that specific charge in the hearing before Pilate in chapter 15?
Likely it originated with the early Christian belief that Jesus was descended from David and is frequently referred to in the canonical gospels as the "son of David," who was the divinely anointed King of Israel (2 Sam 22:51; Ps 18:50). Hence, they gave him the title Christ, that is, "the Anointed One." It is the general view of the New Testament that Jesus was the son of David; that is, he was a descendant of David, the King of Israel (Matt 1:1, 6, 16; 9:7; 12:23; 15:22; 20:30-31; 21:9, 14; 22:42; Luke 1:31-33; 18:37-38; Mark 10:47-48; Rom 1:3; 2 Tim 2:8; Rev 5:5, 22:16). There was apparently a political component to this belief—the disciples are reputed to have believed that he would restore the kingdom to Israel (Acts 1:6; Luke 24:21). The author of the Gospel of John described Jesus as the King of Israel (1:49; 12:13) who was descended from David (7:42) and provides a specific prophecy to that effect (12:12-15).
Oddly, in Mark Jesus is referred to as son of David only twice (10:47-48). That raises the question: would those two appellations be enough to evoke for the general reader the early Christian expectation that Jesus, the Anointed (i.e., the Christ) is the king of Israel in order to account for the heavy emphasis on his kingship in chapter 15? Or is something else going on in Mark? Has Mark, perhaps, deliberately downplayed for political reasons this general early Christian belief in the first fourteen chapters of Mark so as not to raise the ire of Rome?2 The two instances that he is called son of David (10:47-48) would not necessarily raise the ire of Rome, however, for others are also described as a "son of" David, that is, a descendent of David with no inevitable regal expectations (Joseph, for example: Luke 2:4). The two references to Jesus as the son of David might, however, be enough to evoke the early Christian belief in a clandestine way for the knowledgeable reader (cf. Mark 13:14). How do you as a reader of Mark account for Mark's failure to provide in the first fourteen chapters an occasion for this specific charge in chapter 15?
Missouri State University
1G. E. Wright and F. V. Filson, eds., The Westminster Historical Atlas to the Bible (Philadelphia: Westminster Press, 1956), Plate XIV and p. 92. A tetrarch was a ruler over a fourth part of a region.
2Another likely possibility is that Mark was not a careful writer. See Hedrick, "Conceiving the Narrative: Colors in Achilles Tatius and the Gospel of Mark," pages 177-97 in Ronald Hock, et al., eds, Ancient Fiction and Early Christian Narrative (Atlanta, GA: Scholars Press, 1998), 186-97.