The bid activated a release clause in the contract of the 27-year-old, who spent last season on loan at Newcastle, scoring 14 Premier League goals.
He is now expected to discuss personal terms and have a medical with the Reds before completing the move.
Remy was part of the France squad that reached the quarter-finals of the 2014 Fifa World Cup in Brazil and made two appearances as a substitute.
He moved to Newcastle on loan after QPR's relegation from the Premier League in 2013.
The arrival of Remy at Anfield is likely to coincide with the departure of Fabio Borini, after Liverpool accepted a £14m offer for the Italian forward from Sunderland, where he spent last season on loan.
Remy joined QPR during their last top-flight spell, making an £8m switch from French club Marseille in January 2013.
He scored six league goals for the west London club but could not stop them being relegated that season.
QPR regained their Premier League status at the first attempt by beating Derby in the Championship play-off final in May.
Meanwhile, Liverpool defender Andre Wisdom is set to join West Bromwich Albion on loan.
After watching the 21-year-old play in the Reds' 2-1 pre-season friendly win at Preston on Saturday, manager Brendan Rodgers said: "West Brom are an excellent club and and it will be another chance for Andre's education and development to continue."
Wisdom had a successful loan spell at Derby in the Championship last season.
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