Prolific frontman Cox, who scored 32 goals last season, was watched by Albion scouts several times during the 2008/09 campaign following firm interest from former Baggies boss Tony Mowbray.
Mowbray has since joined Celtic, but Albion's interest in Cox has remained and they have now agreed a fee with the Robins for his services.
Swindon chairman Andrew Fitton said: "It is fair to say that we have given West Brom permission to speak to the player.
"I'm not prepared to be drawn on the fee, but I will say that it is not disappointing.
"We have earmarked some of the money to strengthen the squad and you should assume that we are already on the case in finding a replacement."
Cox's agent Alex Cowdy of Touchline Soccer also confirmed the bid.
"I can confirm a bid has been accepted by Swindon from West Brom for Simon and Swindon have now granted us permission to talk to West Brom," Cowdy told skysports.
The deal will stay on ice, however, until Albion chairman Jeremy Peace unveils Mowbray's managerial successor at The Hawthorns.
Preston boss Alan Irvine has now jumped to the top of Peace's wanted list, with the Baggies tipped to make a formal approach for him on Monday.
PNE chairman Derek Shaw has made it clear he wants Irvine to stay at Deepdale, but there is a compensation clause in his contract which allows clubs to talk to him.
Stumping up compensation will not be a problem for the Baggies after they received £2million from Celtic for Mowbray's services.
And Irvine is likely to jump at the chance of taking charge at The Hawthorns, where he would have a much bigger transfer warchest to work with.
If he does step into the Albion hot-seat, his first decision would be whether to sign Cox, who has yet to agree personal terms with the Championship giants.
The 22-year-old has scored 47 goals in 88 appearances for the Robins, and was linked with Celtic while Gordon Strachan was in charge at Parkhead.
Another player who could also move to the Black Country if Irvine takes charge is classy defender Sean St Ledger, who has been linked with a summer move to Albion, Celtic and Burnley.