What a lovely sh-sh-sh quiet whispering sound, meaning, indeed, ‘to whisper’.  A whisper is ‘un chuchotement’, and a whispered conversation is ‘une chuchoterie’. The whisperer, as in horses, is ‘un chuchoteur’ or ‘une chuchoteuse’.

If your whispering is in the line of sweet nothings, see also ‘roucouler’.

For similar onomatopoeia, see also ‘couiner’ and ‘chuinter’.