The forward is out of contract at local rivals Wolves and the Baggies are interested in taking him across the Black Country.
But it is understood Albion are not willing to pay what the Mali international is currently asking.
They are looking for value for money and, while Sako is available on a free transfer, his initial demands would make him one of their top earners.
Albion remain in negotiations with Sako’s camp and they have been made aware of what the Baggies are prepared to pay.
The Premier League side were interested in the 27-year-old in January but Wolves opted not to cash in as they chased promotion.
But Wolves admit they cannot match the wages which Sako could get in the top flight and are braced to lose him with Aston Villa also interested.
Chief executive Jez Moxey said: “Every fan knows the pull of the Premier League, we’re talking about monies now which are so out of the stratosphere that no Championship club can compete with any Premier League club when it comes to a like-for-like financial basis.
“Sako has been a fantastic servant and we’ll be talking to him shortly. We’ll see what he wants to do. We want to keep him but as a free agent when the Premier League comes calling you have to be realistic.”
Sako scored 15 goals in 43 games for Wolves this season as they missed out on the Sky Bet Championship play-offs on goal difference.
He signed for £2.5million from St Etienne in 2012 and has netted 38 goals in 124 games.