Former Cherries manager Pulis guided the Potters into the showpiece final with a resounding 5-0 victory over Bolton Wanderers on Sunday.
The Dorset-based football boss will bid to spring a major surprise when his side take on Manchester City at Wembley next month.
And the Britannia Stadium club will have an extra supporter for the match, with Cherries legend Fletcher keen to see Pulis get his hands on the famous trophy.
He told the Echo: It is going to be very difficult against a top quality Man City side but, in a one-off game, he has got half a chance.
It would be a shock but FA Cup finals are always about shocks.
He will need a bit of luck along the way and Man City to have a bit of an off-day as we know the quality they have got.
I'm sure the bookies will have Man City as favourites and, probably, rightly so.
But in cup games or in any game really, it doesn't work like that.
I would like Stoke to win for Tony. I haven't got any affection with Stoke as a club but because Tony is the manager, I wish him well.
I always want Tony to do well because he did well for me.
Fletcher admitted he owed a lot to Pulis after the then Cherries manager paid £30,000 to take the teenage striker from his hometown club Hartlepool.
Now player-assistant boss and a club legend at Dean Court, Fletcher still remembers meeting Pulis, who quickly persuaded him to make the move south.
Fletcher said: I drove down with my father and it took us about seven hours.
I met Tony on the night and he put me up in the Royal Bath Hotel.
As a Hartlepool lad, to go in there was a bit of an eye-opener for me!
He looked after me and met me the next morning and took me around the training ground at Chapel Gate and really sold the club to me.
He made me feel really welcome.
Within four or five hours, I had decided to move from one end of the country to the other.
I'm indebted to Tony. If it had not been for him, I wouldn't have been at Bournemouth for the past 18 years and I wouldn't have had all the happy times that I have.
If he had not taken a gamble on me buying me from Hartlepool I wouldn't be at Bournemouth now.
Looking back at Pulis's Dean Court reign, which ran between 1992 and 1994, Fletcher added: We did not do brilliantly in the league but we sold the majority of our best players.
When I look back now, I always think we didn't achieve what we wanted under Tony, but he made the club £2million in sales.
When you sell your best players like that, it is so difficult.
With me being in the position I am now, I can understand how hard it was for him.