Well the following verse from Rigved 1.32.9 mentions Vritra as the son of Danu:
8 There as he lies like a bank-bursting river, the waters taking courage flow above him.
The Dragon lies beneath the feet of torrents which Vṛtra with his greatness had encompassed.
9 Then humbled was the strength of Vṛtra's mother: Indra hath cast his deadly bolt against her.
The mother was above, the son was under and like a cow beside her calf lay Danu.
This would make him a Danav (sons of Danu) which is a class of Asurs just like Daityas (sons of Diti). However, the Shatapatha Brahmana 1.6.3 mentions that he was adopted by Danu and is hence called a Danava:
Tvashtri was furious, and exclaimed, 'Has he indeed consumed my Soma uninvited?' However, he himself desecrated the sacrifice, for what pure (Soma) there was left in the tub he let flow (into the fire), saying, 'Grow thou, having Indra for thy foe!' The moment it reached the fire, it developed, or, as some say, it so developed whilst on its way (to the fire). It became possessed of Agni and Soma, of all sciences, all glory, all nourishment, all prosperity.
And since it so developed whilst rolling onwards (vrit), it became Vritra; and since he sprang forth footless, therefore he was a serpent. Danu and Danâyû received him like mother and father, whence they call him Dânava.
So the Danava aspect is explained by this but then SECTION LXVII of Sambhava Parva of Mahabharat while describing all the demons who had incarnated on earth also talks about him as an Asur:
And, O king, that great Asura who was known as Vritra became on earth the royal sage known by the name of Manimat. That Asura who was the younger brother of Vritra and known as Krodhahantri became noted on earth as king Danda. That other Asura who was known by the name Krodhavardhana became noted on earth as the monarch, Dandadhara. The eight sons of the Kaleyas that were born on earth all became great kings endued with the prowess of tigers. The eldest of them all became king Jayatsena in Magadha. The second of them, in prowess, like Indra, became noted on earth as Aparajita. The third of them, endued with great energy and power of producing deception, was born on earth as the king of the Nishadas
It is interesting that even though he was called an Asura he was reborn as a Muni. The Bhagvat Puran also mentions him as a great devotee of Vishnu. But in Chapter XVI of Devi Bhagwat Puran again he gets mentioned as an Asur:
Again in Book 4 Chapter IX it is mentioned:
Others have given different references in their answers so I will mention a different indirect reference from the Varah Puran. In Varah Puran 28.2-15 the rebirth of Vritra is mentioned in which he sires an Asur son called Vetra who again torments the Devas and is finally killed by the goddess:
5 In his former birth, he was the son of Tvasta, and being
immune against the attack of any weapon, he was killed with
foam. 6-7. Killed by foam, he was dissolved in it. Then he was
born in Brahma’s line as the valiant Sindhudvipa, and he per-
formed penance always thinking of his enmity to Indra.
8.After a long time, the auspicious river Vetravati assumed
a beautiful human form and went to the place where the king
was engaged in his austerities.
13 In Vetravati was born immediately a son named Vetra who was resplendent like the dozen suns, mighty and brilliant, and he became the king of Pragjyotisa. 14-15. In due course he became a powerful and courageous
youth, much lucky, and he conquered the whole world consisting of seven islands and then, going up the Meru mountain, conquered Indra, then Agni and then Yama. 16. Indra, when vanquished, went to Agni, Agni when vanquished went to Yama, Yama when vanquished went to Niirti, and Nirrti when vanquished went to Varuna. 17. Varuna with Indra and others went to Vayu, and Vayu went to Kubera with them. 18. Kubera went to his friend Siva. The Asura was chasing
them all with his club, and so he too came to Sivaloka.
19 Siva, knowing that he could not be easily killed, took
the gods to Brahma worshipped by Siddhas and others. 20-21 . There Brahma, having created the world, was engaged in meditating on Gayatri, the Maya of the Supreme Being, within the water which was flowing from Visnu’s feet. At this Devas cried aloud: 22. Brahma, protect all of us Devas and sages, who have
now come to you being harassed by the demon (Vrtra)”.
23 Prayed thus, Brahma looked at Devas and contemplated on Maya by whose prowess there will be no Asuras or Raksasas (left to harass them). 24-26. Then appeared there out of the water the great Goddess, a maiden wearing garlands and crown, clad in white cloth,with eight hands each holding a divine weapon, disc, conch, mace, rope, sword, bell, bow (and arrow) and with a quiver,
riding on a lion.
27 She fought with the Asuras with great agility for a thou-
sand divine years and with various divine weapons.
28 Finally the demon Vetrasura was killed by her in battle
and there was then a great shout of joy in the ranks of Devas.
So in conclusion, even if he was a pious soul, it would be Vritra's actions of attacking the Devas that would probably lead him to being called an Asur.