Is it permissible for someone who missed a prayer due to forgetfulness or sleep to make it up immediately upon remembering it, upon hearing the Adhan for the next prayer, before its iqamah, and is there any basis for this in the Qur’an or Sunnah?

If one sleeps through a prayer or forgets it, he must pray it as soon as he remembers it, because the Prophet, peace and blessings be upon him, said: "If any of you oversleeps or is heedless of a prayer, he must pray it when he remembers it," and because Allah Almighty said: "{Establish prayer for My remembrance}."