HITC Sport understands that they want player plus more cash from Boro.
The Scottish international is regarded as one of the best players outside the Premier League and a host of top-flight clubs have been linked to him since their relegation from the top-flight.
West Bromwich Albion were believed to be lining up a bid for him this month, but now it is Boro have come in with an offer for the 24-year-old winger.
Sources close to QPR have confirmed to HITC that a bid has come in, but it has been rejected - but they have told Boro that they would accepted a slightly increased offer and also a player, believed to be the Teessiders young winger Adam Reach - who is currently on loan at Preston.
Should Boro come in with an offer with Reach included, then HITC Sport has been told that it would be accepted, as whilst Jimmy Floyd Hasselbaink would love to keep him - he has been promised the majority of the cash to reinvest in his squad.
Albert Adomah, a QPR fan, has long been linked to Loftus Road by West London Sport and could be seen as another target, but it appears unlikely Boro would be willing to sell him at this point.
Boro's move for Phillips does not impact on their efforts to sign Fulham captain Ross McCormack and if they were able to land the pair, it would send a huge statement of intent to their Championship promotion rivals.