The birth of Hanuman is mentioned in different Puranas. Different places are identified as the birthplace of Hanuman.

(a)
Valmiki Ramayana: While Jambavanta awakened the strength of Hanuman to make him cross the ocean, Jambavanta told about the birth of Hanuman. Punjikasthala, an apsara, had to take birth on earth as a vanara due to a curse. She took birth as Anjana, the daughter of Kunjara and the wife of Kesari. She could change her form at will. Once, assuming a human physique, and decorating herself richly with silken clothes, she moved about on the top of a mountain. The Wind-god saw her and became attracted by her beauty. The Wind-god invisibly embraced and impregnated her. The Wind-god told Anjana that she will bear a child of great prowess equal to him in agility. Anjana gave birth to Hanuman in a cave.
(Valmiki Ramayana - Kishkindha Kanda - 66)
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Anegundi Anjanadri (Koppal district, Karnataka)

(b)
Skanda Purana - Venkatachala Mahatmya: Anjana performed penance in Vidyaganga at Venkatachala (on the advice of sage Matanga) to get a son. Anjana performed penance with the Prana-Vayu (Vital Breath) as her object of worship. First she subsisted on fruits, then she subsisted on water and finally she did not eat anything. She performed penance with her eyes fixed on the tip of her nose. Her friend Vipula rendered service to her. Anjana performed penance for thousand years. Then the Wind-god appeared before her (during Sun in Aries, full-moon, constellation Chitra) and asked her to choose a boon. When she asked for a son, the Wind-god said that he himself will become his son. Then Brahma, Vishnu, Shiva and various devas and sages came there. Then sage Vyasa told that a herioc son will be born to her.
(Skanda Purana - Venkatachala Mahatmya - 39 to 40)
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Tirumala Anjanadri (Tirupati district, Andhra Pradesh)

(c)
Brahma Purana - Gautami Mahatmya: Anjana mountain is located at the side of Brahmagiri mountain on the southern bank of river Gautami Ganga (Godavari). On that mountain stayed celestial ladies Anjana and Adrika who had their downfall due to a curse. Due to the curse, Anjana's face resembled that of a monkey and Adrika's face resembled that of a cat. They were wives of Kesari. Once Kesari went to the Southern Sea. Sage Agastya arrived there. Both of them worshipped the sage duly. Sage Agastya became pleased and told them to ask for a boon. Both of them asked for a mighty son. Sage Agastya granted the boon and went to the southern side. Once when Anjana and Adrika were sporting on the top of the mountain, Vayu deva and Nirrita devas came there. They were attracted towards Anjana and Adri and asked them to become their wives. Anjana and Adri accepted them. They sported on the top of the mountain. Hanuman was born to Anjana and Vayu deva and Adri was born to Adrika and Nirriti.
(Brahma Purana - Gautami Mahatmya - 14)
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Nashik Anjaneri (Nashik district, Maharashtra)