The forward enjoyed his best season in the Premier League last term, scoring 19 times in the top flight.
However, he is now being linked with a host of top teams who are seeking a first class center forward.
Anderton, who played at Euro 96 and the 98 World Cup with England, told the Daily Echo: “He’s obviously in a place that he’s happy and that he’s able to be talked about as playing for England and that sort of thing.
“So if he’s doing the business, it’s not like Gareth is only looking at the so-called big clubs in order to pick his players.
“I think if he’s happy there, that’s where he should stay.
“Bournemouth are on the up, they’re improving every year. I would be very surprised if he would be looking to get away.
“Of course, I understand having spent 12 years of my career at Tottenham, that if that were to happen (interest from Spurs), I’m sure his head would be turned.
“But, at the same time, I don’t think it’s something he will be pursuing himself.
“If that were to happen, I’m sure he would go and have a conversation with whoever it may be.”