The Quran

وَإِذ قالَ عيسَى ابنُ مَريَمَ يا بَني إِسرائيلَ إِنّي رَسولُ اللَّهِ إِلَيكُم مُصَدِّقًا لِما بَينَ يَدَيَّ مِنَ التَّوراةِ وَمُبَشِّرًا بِرَسولٍ يَأتي مِن بَعدِي اسمُهُ أَحمَدُ ۖ فَلَمّا جاءَهُم بِالبَيِّناتِ قالوا هٰذا سِحرٌ مُبينٌ

Ali Unal

And Jesus son of Mary said: “O Children of Israel! Surely I am the Messenger of God sent to you, confirming (whatever of the truth is contained in) the Torah which was revealed before me,¹ and bringing the glad tidings of a Messenger to come after me, whose name is Ahmad.”² But when he came to them (the whole of humankind including the later generations of the Children of Israel) with the manifest signs (of his being God’s Messenger), they said: “This (which he preaches and does) is clearly (nothing but) sorcery.”

Jesus, upon him be peace, did not come with a new religion or message from God. He communicated the same message as all the previous Prophets had done, and the Sharī’ah of Moses, upon him be peace; the only difference was that he made lawful for them certain things that had been forbidden to them (3: 50). He also made clear to them some of the matters in which they differed (43: 63). For Jesus, upon him be peace, and other dimensions of his Messengership, see sūrah 3: 48–51, notes 8–9; sūrah 5: 46–47, 110, 116–118; sūrah 19: 30–34.
Every Prophet gave the glad tidings of the Messenger to come after him and declared his faith in him. (For the mission of the Messengers and the preceding Messenger’s giving the tiding of the advent of the one to follow, see 3: 81, note 15; and for why God sent a Messenger with a Book after the others, see 5: 48, note 11.)
So, it is natural that Jesus gave the glad tidings of the Prophet Muhammad, upon them be peace. It is also clear in the Gospel of John that the Children of Israel had been expecting another Prophet besides Jesus, upon him be peace:
Now this is the testimony of John, when the Jews sent priests and Levites from Jerusalem to ask him, “Who are you?” He confessed, and did not deny; but confessed, “I am not the Christ.” And they asked him, “What then? Are you Elijah?” He said, “I am not.” “Are you that Prophet?” And he answered, “No.” Then said they to him, “Who are you, that we may give an answer to those who sent us? What do you say about yourself?” He said: “I am the voice of one crying in the wilderness: ‘Make straight the way of the Lord,’ as the prophet Isaiah said.” (John, 1: 19–23)
As understood from this passage, the Children of Israel had been expecting the coming of the Christ (Messiah), Elijah, upon him be peace, and another Prophet (that prophet) who must have been known and who was expected by everyone at that time.
As explained in Appendix 1, this prediction is supported by several references. Paráklētos (differently rendered as Counselor, Helper, or Comforter in different versions of the New Testament) is, in fact, a corruption of Períklytos (the “Much-Praised”). Its Aramiac counterpart is Mawhamana, which means Ahmad. Ahmad and Muhammad are derived from the same root verb, “Ha-Mi-Da” meaning to praise. However, Ahmad also means one who praises. In many Prophetic Traditions, it is stated that one of the Prophet Muhammad’s names is Ahmad, and this is mentioned by Hasan ibn Thābit, a famous poet during the Prophet’s time, in one of his poems. So the Prophet Muhammad, upon him be peace, was also known and mentioned as Ahmad during his own lifetime. Though the name Ahmad had not been known and used before him among the Arabs, this name became widely known and used during his time. It is of interest that Bediüzzaman Said Nursi records that the Prophet Muhammad, upon him be peace, was mentioned in the Torah also with the name Munhamanna, meaning Muhammad, the praised one. (For other predictions of the Prophet Muhammad in the Old and New Testaments, see Appendix 1.)
Concerning the Prophet Muhammad’s mission regarding the People of the Book, and their relationship with him, the Qur’ān states: They follow the (most illustrious) Messenger, the unlettered Prophet, whom they find described (with all his distinguishing features) in the Torah and the Gospel (that are) with them. He enjoins upon them what is right and good and forbids them what is evil; he makes pure, wholesome things lawful for them, and bad, corrupt things unlawful. And he relieves them of their burdens (remaining of their own Law) and the restraints that were upon them. So those who believe in him (with all sincerity), honor and support him, and help him, and follow the Light (the Qur’ān) which has been sent down with him – those, they are the truly prosperous (7: 157).