The Gunners were linked with a surprise swoop for the 28-year-old after Per Mertesacker and Gabriel Paulista picked up injuries in pre-season.
That forced Arsene Wenger to select youngsters Calum Chambers and Rob Holding during a 4-3 defeat to Liverpool at the Emirates last Sunday.
The north London club have continued to be linked with the former Manchester United defender, who left Old Trafford for the Hawthorns last summer.
The northern Ireland international, who featured prominently for his country at Euro 2016, was a key part of the Baggies side that finished 14th last season and impressed on the opening weekend of the 2016/17 Premier League campaign with a strong performance at left-back against Crystal Palace.
And Baggies gaffer Tony Pulis has suggested the club are open to agreeing an Arsenal transfer for the centre-back, but only if they sign a replacement first.
As quoted by The Daily Mail, he said; “You don’t want to lose your best players, but in football sometimes that happens. If it does happen, you need to make sure you know where your replacements are.
“If Arsenal come in for Jonny, if we do any business we’ll do it between the clubs. I wouldn’t talk here about doing a deal with Arsenal or anybody else.
“The most important thing is that Jonny is with us and we want him to stay because he’s an outstanding player.
“Last Saturday, whether it was Townsend or Zaha, and it’s not Jonny’s favourite position playing as a full back, he was most probably our best player on the pitch.”