According to The Mirror, Stoke will have to pay West Brom a transfer fee of £16 million for the services of Berahino.
The former England Under-21 international is a promising young striker, and a number of clubs had been credited with interest in him.
According to The Mirror, Crystal Palace, Watford and Everton have been tracking Berahino.
The same newspaper reported earlier this month that West Brom were ready to offer the youngster to West Ham United in a swap deal for Diafra Sakho.
The Guardian recently reported that Tottenham Hotspur wanted to sign Berahino from West Brom in the summer transfer window.
Berahino scored four goals and created 18 chances in 31 Premier League appearances for West Brom during the 2015-16 campaign.
The youngster had a pass accuracy of 78% and a shot accuracy of 44%, and won eight headers, 18 dribbles and 14 tackles.