The Sun is reporting that the Reds are interested in a deal to bring the 25-year-old to Anfield this summer.
According to the same story, Fraser has the offer of a new contract on the table at Bournemouth, but the Scot looks set to turn it down.
Fraser has one year left on his current contract and is attracting interest from Arsenal and Manchester United this summer, according to the same report.
The same article says that Bournemouth do not want to sell him this summer but know that they risk losing him for nothing next summer, when his current deal is set to expire.
It is also claimed in the same story that a bid in the region of £30m would be enough to tempt Bournemouth into selling Fraser this summer.
Fraser was in good form for Bournemouth this season, scoring seven goals and making 14 assists in 38 Premier League appearances for the Cherries.
Liverpool FC, meanwhile, are currently preparing for their Champions League final showdown against Tottenham Hotspur in Madrid on 1 June.
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