Barry, who has also been courted by Barcelona this summer, turned 16 last Friday and is now vulnerable to being poached by European clubs.
Albion have offered the England under-16 striker the equivalent of a three-year deal, a one-year scholarship followed by two-year professional contract when he turns 17.
And Dowling believes the lethal forward from Sutton Coldfield will stay.
“Louie is one of our young players that has got a three year contract offered to him that we’re confident he’s going to sign,” said Dowling.
“The message we’re getting back is he will sign. He’s a local boy.”
However, PSG are the latest club to show interest in Barry, and are believed to have made him an offer.
The youngster shot to prominence thanks to his exploits for England’s youth teams.
He scored ten goals in five games in a tournament for England under-15s last year, and won the golden boot at a tournament for England under-16s in November.
If Barry leaves for an overseas club, the Baggies will only receive Fifa’s flat rate for compensation of £235,000.