The Baggies are set to pay Rangers in the region of £1.5million for the 23-year-old, who will bolster Slaven Bilic's back-line.
Furlong's arrival will allow Allan Nyom to be sold to Spanish club Getafe, and plug the gap at right-back.
Mason Holgate has returned to parent club Everton, although the Baggies would be keen to bring him back on loan.
However, they are waiting to see if the Merseyside club are willing to let Holgate leave again.
Signing Furlong would not cool their interest in Holgate, who can play centre-back as well as right-back.
Furlong is the son of former QPR striker Paul Furlong, and played 20 times in the Championship for Rangers last season.
He has previously had loan spells at Northampton Town, Cambridge United and Swindon Town.
The Baggies are starting to make significant moves in the transfer window, having signed Kenneth Zohore and Semi Ajayi on Friday and Saturday respectively.
They retain an interest in Sporting winger Matheus Pereira, and are keen to bring him in on a season-long loan with an option to buy.
Negotiations with the Portuguese club remain ongoing, but the Baggies are exploring a number of attacking options.
They are also interested in former academy player Romaine Sawyers, but are one of many clubs looking at the Brentford playmaker.