The west London club are keen to sign the Championship defender, according to various reports, with manager Harry Redknapp said to be a fan of his after seeing him in action first-hand last term.
Sky Sports are reporting that the club have made an undisclosed bid believed to be around £4m, while also pointing out that Everton were fined £45,000 by the Football Association for making an unauthorised approach for the Nottingham Forest man four years ago.
In order to steal a march on their rivals, QPR are again quoted as having bid the £4m figure for the 20-year-old, for whom Arsenal reportedly had a bid rejected in the past.
Lascelles played 38 times last season in the Championship and has represented England at Under-18, U19 and U20 levels.
Should the Forest defender make the move to Redknapp's side, he could be accompanied by experienced duo Rio Ferdinand and Brede Hangeland, who have both been heavily linked with the west London side.
QPR are also in negotiations with Armand Traore, whose contract expires this summer but Redknapp is keen to keep at the club.
Nottingham Forest are said to want to keep Lascelles at the City Ground, however, with the 20-year-old also previously expressing his admiration for coach Stuart Pearce.
"I am really looking forward to it, to working under him," Lascelles said of the new Forest boss back in April.
"I have met him once and he seemed like a really nice guy. He is a fans' favourite and, if you can't learn off somebody like him, then you can't learn off anyone.
"I think he will be ideal for me. He was a defender, so he will be the ideal person to learn from."