The Arabic word وسوس (waswas) in various forms appears 5 times in the Qur'an.
The first two refer to the story of Adam and Eve. The fa- prefix means "but" or "then".
But the Shaitan
made an evil suggestion (فَوَسْوَسَ,
fa-waswasa) to them that he might make manifest to them what had been hidden from them of their evil inclinations, and he said: Your Lord has not forbidden you this tree except that you may not both become two angels or that you may (not) become of the immortals.
But the Shaitan
made an evil suggestion (فَوَسْوَسَ,
fa-waswasa) to him; he said: O Adam! Shall I guide you to the tree of immortality and a kingdom which decays not?
Then one time it's a persons soul which whispers. The tu- simply indicates present tense, 3rd person, feminine, singular verb.
And We have already created man and know what his soul
whispers (تُوَسْوِسُ,
tuwaswisu) to him, and We are
closer to him than [his] jugular vein
And two times in a short sura 114. The al- indicates definite article (before a noun). The yu- indicates present tense, 3rd person, singular, masculine verb.
114:1 Say: I seek refuge in the Lord of men,
114:2 The King of men,
114:3 The god of men,
114:4 From the evil of the retreating whisperer (ٱلْوَسْوَاسِ, al-waswasi)
114:5 Who whispers (يُوَسْوِسُ, yuwaswisu) into the hearts of men,
114:6 From among the jinn and the men.
Tafsir Al-Jalalayn says that the "whisperer" in 114:4 is Satan:
from the evil of
the slinking whisperer Satan — he is referred to by the name of the action waswasa on account of his repeated engaging in it — who slinks away and recoils from the heart whenever God is mentioned
Tafsir Al-Jalalayn on 114:4
From this we can conclude that the concept of waswas (Satan whispering into Muslim minds) is described in the Qur'an, so it is definitely a part of the Islamic doctrine.
Muslims are forgiven thoughts against Islam (the "whispers from Satan") as long don't act upon them and keep silent:
Narrated Abu Huraira:
The Prophet (ﷺ) said, "Allah has accepted my invocation to forgive what
whispers (وَسْوَسَتْ,
waswasat) in the hearts of my followers, unless they put it to action or utter it."
Narrated Abu Huraira:
The Prophet (ﷺ) said, "Allah forgives my followers those (evil deeds) their souls may
whisper (وَسْوَسَتْ,
waswasat) or suggest to them as long as they do not act (on it) or speak."
It was narrated that Abu Hurairah said:
"The Messenger of Allah said: 'Allah, the Mighty and Sublime, has forgiven my Ummah for what is
whispered (وَسْوَسَتْ) to them or what enters their minds, so long as they do not act upon it or speak of it.'"