Akinola, who can also play up front, only moved to Barnet in January from National League side Braintree, but has scored two goals and created as many in 19 Bees appearances.
General view of The Hive, home of Barnet FC
And The Mirror reports the League Two side are looking for £100,000 for the 24-year-old Nigerian, formerly of Harrow Borough and Boreham Wood.
Rangers were linked with a move for another winger of Nigerian descent, Fred Onyedinma, earlier in the summer, but the Millwall player signed a new deal instead.
QPR boss Ian Holloway has struggled to add to his squad this transfer window, with Josh Scowen the R's only recruit of note so far.
QPR manager Ian Holloway
Charlton, on the other hand, have signed two wingers already in Tarique Fosu and Mark Marshall, and kept hold of wanted wide man Ricky Holmes.
Should Queens Park Rangers / Charlton Athletic move for the Barnet winger?