Arsenal have identified Fraser as a potential in at the Emirates during this transfer window.
The Scottish international made 14 assists last season – one behind the league high total set by Eden Hazard.
His exploits on the south coast have attracted interest and the Gunners look most likely to pounce.
Bournemouth though are demanding £30m for the 25-year-old and Fraser believes that price tag means he will be staying put.
“I don’t think anyone will be paying £30m,” he told Goal.
“If something happens, it’ll happen. If it doesn’t, I’m not one to throw my rattle out of the pram. I’ve got a year left with Bournemouth.
“If I’m there, I’ll give my all. If I’m somewhere else, I’ll give my all.”
Fraser moved to the Vitality Stadium six years ago when the Cherries were still in League One and has risen through the divisions with them.
His stock now couldn’t be higher having made his Scotland debut two years ago.
Arsenal are looking to add in a variety of areas and Unai Emery had seen Fraser as a worthy addition.
Top six clubs have previously had their pick of the talent elsewhere in the Premier League, but an increase in financial power among the perceived lesser clubs has seen them stand firm.
The Gunners are also tracking Belgian winger Yannick Carrasco who is believed to want out of China.
He moved to the lucrative Chinese Super League from Atletico Madrid in 2018, but is seeking a return to Europe.
Carrasco scored for Atletico in the 2016 Champions League final and was viewed very highly, but there are concerns that his level will have dropped in Asia.