Can a Muslim be a sincere friend to a kaafir?

It is not permissible for a Muslim to make friends with a mushrik or to take him as a close friend, because Islam calls on us to forsake the kaafirs and to disavow them, because they worship someone other than Allaah. Allaah says (interpretation of the meaning): 

“O you who believe! Take not as friends the people who incurred the Wrath of Allaah (i.e. the Jews). Surely, they have despaired of (receiving any good in) the Hereafter, just as the disbelievers have despaired of those (buried) in graves (that they will not be resurrected on the Day of Resurrection)”

[al-Mumtahanah 60:13]

This was also the teaching of the Prophet (peace and blessings of Allaah be upon him). 

1 – It was narrated from Abu Sa’eed al-Khudri (may Allaah be pleased with him) that he heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say, “Do not keep company with anyone but a believer and do not let anyone eat your food but one who is pious.” (Narrated by al-Tirmidhi, 2395; Abu Dawood, 4832. Abu ‘Eesa al-Tirmidhi said: this hadeeth is hasan. It was also classed as hasan by al-Albaani in Saheeh al-Tirmidhi, 2519). 

Abu ‘Eesa al-Khattaabi said: Rather he warned against keeping company with anyone who is not pious and against mixing with them or eating with them, because eating with a person instills friendship and love in the heart. 

He said: do not make friends with anyone who is not pious; do not take him as a companion with whom you eat and chat. 

(Ma’aalim al-Sunan, Haamish Mukhtasar Sunan Abi Dawood, 7/185, 186). 

2 – It was narrated from Samurah that the Prophet (peace and blessings of Allaah be upon him) said: “Do not live among the mushrikeen and do not mix with them, for whoever lives among them or mixes with them is not one of us.” (Narrated by al-Bayhaqi, 9/142; al-Haakim, 2/154. He said, it is saheeh according to the conditions of al-Bukhaari. The hadeeth was also classed as saheeh by al-Albaani in al-Silsilat al-Saheehah, 2/229 with its corroborating reports). 

But it is permissible to deal with them in a kind manner in the hope that they might become Muslim. 

It was narrated that Anas (may Allaah be pleased with him) said: There was a Jewish boy who used to serve the Prophet (peace and blessings of Allaah be upon him), and he fell sick. The Prophet (peace and blessings of Allaah be upon him) came to visit him. He sat by his head and said, “Become Muslim.” (The boy) looked at his father who was with him, and he (the father) said, “Obey Abu’l-Qaasim (peace and blessings of Allaah be upon him).” So he became Muslim, and the Prophet (peace and blessings of Allaah be upon him) went out, saying, “Praise be to Allaah Who has saved him from the Fire” (Narrated by al-Bukhaari, 1290).

 And Allaah knows best.