The Super Hoops boss has made reinforcing his backline a priority this January, with the London club having conceded 36 goals in 21 Premier League games, leaving them at the foot of the table.
His defensive crisis has worsened recently with the news that experienced centre-half Ryan Nelsen will leave to take over as manager of MLS side Toronto.
‘Arry believes that Olsson is the perfect man to slot into his squad and has pressed ahead with a £5m bid.
The Swede will be available for £4m during the summer, due to a clause in his contract, but QPR hope that the Baggies will decide to cash in now to avoid losing him at a reduced fee.
One problem may be persuading the 29-year-old to swap potential European qualification at The Hawthorns for a relegation fight in West London.
Although Redknapp will be able to offer him an attractive contract, the former Spurs boss will be reluctant to allocate all of his budget to one signing, with the club looking to reduce their spending.
As well as Olsson, QPR are said to be looking at Tottenham’s William Gallas, who could be available this winter.