Solanke's arrival, likely to be the first of two moves from Anfield to the Vitality Stadium, with defender Nathaniel Clyne set to join on loan until the end of the season, will pave the way for Jermain Defoe to join Rangers on an 18-month loan.
Solanke began his career at Chelsea, playing a season on loan at Dutch side Vitesse Arnhem before joining Liverpool in 2017.
He made 27 appearances for Liverpool in his first season but has not featured for Jurgen Klopp's side in the current campaign.
The 21-year-old is keen to play first-team football, having left Chelsea for exactly the same reason and with little chance of displacing the trio of Roberto Firmino, Sadio Mane or Mohamed Salah.
"I think the club (Bournemouth) is going somewhere," Solanke told afcbTV. "Over the past few years, they have been really good in the Premier League and had some good positional finishes as well. I think it's a good club to be at, especially for someone of my age."
Bournemouth manager Eddie Howe said: "Dominic is technically gifted, very athletic and has the physical profile that will give us something different in our front line."