The 23-year-old, who has won praise for her whip-cracking, witty wordplay, hails from Croydon and is the cousin of grime artist Stormzy.
The annual award aims to showcase the best new artists of the coming year, and is voted for by 170 DJs, music critics, radio producers and writers.
"That is epic!" said Rose on learning she had made the top five.
"Being on the longlist was an honour. This is just... wow. I really wasn't expecting that."
Rose, who began making music in her spare time while working in a betting shop, has scored viral hits with songs like Skwod and Tight Up.
She describes her sound as "really bassy, full of fun and full of energy".
The Sound of... list launched in 2003 and has a track record in championing new and innovative acts, including Adele, Sam Smith, Ellie Goulding, Frank Ocean and Florence + The Machine.
This year's top five will be revealed on BBC Radio 1 and online throughout the week, with the winner announced on Friday.
Last year's victor was Jack Garratt, who praised the "very diverse" range of acts on the 2017 list, and encouraged fans to embrace new talent.
"There's more music out there than just your favourite act," he said. "It all deserves respect and it all deserves love.
"Even though there are 15 names on the longlist, there are hundreds more that definitely could have been. I think 2017 is going to be a very interesting year."
According to the rules, the on the longlist acts should not be well known - so talent show contestants and members of famous bands launching a solo career are excluded.
Artists must not have had a UK top 20 single or album before 30 October 2016 - although guest vocalists remain eligible.
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South London rap prodigy Nadia Rose has taken fifth place on the BBC Music Sound of 2017 list.