Short answer
In the Quran's sense, Jesus was a muslim: one who submits to God, as every prophet was, and his disciples said "we are Muslims". By people and community he was an Israelite, sent to the Children of Israel and confirming the Torah. Muslims do not claim he followed the later law given to Muhammad ﷺ during his first life on earth.
Muslims believe all prophets brought one religion of submission to Allah (islam), with different laws for different peoples. Isa confirmed the Torah, with some changes Allah revealed to him. When he returns, he will rule by the shariah of Muhammad ﷺ.
The four schools agree on this.
Explanation
The Arabic word islam means submission to Allah, and muslim means one who submits. In that sense the Quran describes every prophet and his true followers as muslims. It says Abraham "was neither a Jew nor a Christian, but he was one inclining toward truth, a Muslim". When Jesus asked who would support him, his disciples answered: "We have believed in Allah and testify that we are Muslims." The Prophet ﷺ said the prophets are brothers whose religion is one.
At the same time, Jesus lived among his own people. The Quran says he was sent as "a messenger to the Children of Israel", confirming the Torah before him and making lawful some things that had been forbidden to them. He worshipped the God of Abraham, prayed and fasted. The Gospels themselves describe him falling on his face in prayer, a posture Muslims recognise from their own sujud (prostration).
- Was he Jewish? He was an Israelite who lived under the law of Moses. Muslims usually say Israelite prophet rather than using later labels.
- Was he Christian? The word came later: the Book of Acts says the disciples were first called Christians in Antioch, after Jesus had been raised. The Quran calls his followers al-Nasara.
- After his return: he will follow and rule by the law of Islam, as a member of this ummah. See will Jesus return?
So Muslims see Jesus as part of one unbroken line of prophets calling to the same faith. See is Jesus a prophet or messenger? and Prophet Isa.
The evidence
Quran, Surah Aal-i-Imraan 3:52
۞ فَلَمَّآ أَحَسَّ عِيسَىٰ مِنْهُمُ ٱلْكُفْرَ قَالَ مَنْ أَنصَارِىٓ إِلَى ٱللَّهِ ۖ قَالَ ٱلْحَوَارِيُّونَ نَحْنُ أَنصَارُ ٱللَّهِ ءَامَنَّا بِٱللَّهِ وَٱشْهَدْ بِأَنَّا مُسْلِمُونَ ﴿٥٢﴾
But when Jesus felt [persistence in] disbelief from them, he said, "Who are my supporters for [the cause of] Allah?" The disciples said, "We are supporters for Allah. We have believed in Allah and testify that we are Muslims [submitting to Him].
The disciples of Jesus called themselves Muslims, those who submit.
Quran, Surah Aal-i-Imraan 3:67
مَا كَانَ إِبْرَٰهِيمُ يَهُودِيًّۭا وَلَا نَصْرَانِيًّۭا وَلَٰكِن كَانَ حَنِيفًۭا مُّسْلِمًۭا وَمَا كَانَ مِنَ ٱلْمُشْرِكِينَ ﴿٦٧﴾
Abraham was neither a Jew nor a Christian, but he was one inclining toward truth, a Muslim [submitting to Allah]. And he was not of the polytheists.
The prophets' religion is submission to Allah, beyond later labels.
Quran, Surah Aal-i-Imraan 3:50
وَمُصَدِّقًۭا لِّمَا بَيْنَ يَدَىَّ مِنَ ٱلتَّوْرَىٰةِ وَلِأُحِلَّ لَكُم بَعْضَ ٱلَّذِى حُرِّمَ عَلَيْكُمْ ۚ وَجِئْتُكُم بِـَٔايَةٍۢ مِّن رَّبِّكُمْ فَٱتَّقُوا۟ ٱللَّهَ وَأَطِيعُونِ ﴿٥٠﴾
And [I have come] confirming what was before me of the Torah and to make lawful for you some of what was forbidden to you. And I have come to you with a sign from your Lord, so fear Allah and obey me.
Jesus confirmed the Torah given to his people, with some changes.
Quran, Surah Aal-i-Imraan 3:19
إِنَّ ٱلدِّينَ عِندَ ٱللَّهِ ٱلْإِسْلَٰمُ ۗ وَمَا ٱخْتَلَفَ ٱلَّذِينَ أُوتُوا۟ ٱلْكِتَٰبَ إِلَّا مِنۢ بَعْدِ مَا جَآءَهُمُ ٱلْعِلْمُ بَغْيًۢا بَيْنَهُمْ ۗ وَمَن يَكْفُرْ بِـَٔايَٰتِ ٱللَّهِ فَإِنَّ ٱللَّهَ سَرِيعُ ٱلْحِسَابِ ﴿١٩﴾
Indeed, the religion in the sight of Allah is Islam. And those who were given the Scripture did not differ except after knowledge had come to them - out of jealous animosity between themselves. And whoever disbelieves in the verses of Allah, then indeed, Allah is swift in [taking] account.
Submission to Allah is the one religion of all the prophets.
Sahih al-Bukhari 3443
Sahih... أمهاتهم شتى ودينهم واحد ...
"Their mothers are different, and their religion is one."
All the prophets, Jesus included, shared one religion.
Where the ruling is recorded
- al-Fiqh al-Akbar, the section on the prophets. Belief in all the prophets as bearers of the same faith.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.