Bartley, who has also agreed personal terms with the club, is set to finalise his move following a medical today.
The 27-year-old centre-back will bolster Albion’s options at the back following the exits of Gareth McAuley and Jonny Evans.
Although his signing could be confirmed before tomorrow’s pre-season trip to Swindon Town, he is unlikely to feature in the game, having not yet trained with the Baggies.
Bartley will become Albion’s second signing of the summer following their £6.5m capture of Manchester United goalkeeper Sam Johnstone.
Albion have beaten Championship rivals Leeds United to the defender’s signature, while Birmingham City were also interested.
Bartley was a Bolton youth team player before Arsenal signed him at the age of 16. He won the FA Youth Cup with the Gunners before joining Swansea in 2012 for £1m.
He struggled to become a regular in South Wales, and made just 38 appearances for Swansea during six years.
He has, however, enjoyed successful loan spells in the Championship at both Birmingham and Leeds in the past few years.
The Baggies are in the process of replacing the players that have already left the club this summer, and head coach Darren Moore has confirmed there are offers in place for one or two more players.
Although Albion do have parachute payments to help them in the transfer market, the Baggies are currently operating in an overdraft thanks to last summer’s £40m splurge.
The club is unable to splash out on significant transfer fees without receiving money from sales.
Albion have received just over £7m so far this summer for Jonny Evans and Ben Foster, and spent £10.5m on Johnstone and Bartley.
However, they could still be given a cash injection later in the window that will supplement more signings.
Nacer Chadli has a £17m release clause, Salomon Rondon has a £16.5m release clause, and Ahmed Hegazi has a £10m release clause.
The clauses time out on different days in late July and early August, giving the club time to buy replacements before the self-imposed transfer deadline for English clubs on August 9.
After the clauses time out, the Baggies do not have to accept bids of those figures for those players.
Albion could also get significant money for Craig Dawson and Jay Rodriguez, who they value at £20m each.
Burnley are interested in both players, but are nowhere near Albion’s asking price for either at the moment.