Newcastle and Tottenham, who had several offers for Berahino rejected in the summer, are thought to be interested in signing the Albion academy product, with a number of newspapers reporting on Sunday that the Magpies had offered £21 million for the player.
Spurs made an offer for the England Under-21 international in the summer and the forward was keen to move - infamously expressing his disappointment at not being allowed to leave on Twitter - but are yet to return for the player this month. Newcastle hope they can take advantage of that and make an offer that finally entices the Baggies to sell.
The Press Association understands Peace will not sell unless a sizeable bid comes in and it will be interesting to see if a club stumps up such an amount before the January window closes at 11pm on Monday.
Asked about Berahino's future on Saturday, Pulis said: "There's two more days to wait. The chairman will make the decisions on the financial matters and that's really what it boils down to.
"I think football-wise, everyone would want to keep Saido here. When you put Saido in front of goal, he's a goalscorer.
"But we need to get him fit because he's not fully fit. Even today (Saturday), he scored two goals, two great goals, he took them fantastically well, but he's not as fit as he should be.
"He accepts that and we certainly know that he's not like he was last year. But having said all that, we could certainly do with him staying - because we'll need his goals.
"He has that wonderful, wonderful ability to put the ball in the back of the net."
Meanwhile, West Brom have confirmed that Arsenal midfielder Serge Gnabry has returned to his parent club after his stay at The Hawthorns was terminated.