The police said they had launched a preliminary investigation after the tabloid BT said it had seen evidence of leaks from the Danish unit of payment systems provider Nets to Se & Hoer, Scandinavia's largest celebrity publication.
"We read in national newspaper BT that information had been leaked from Nets to Se & Hoer and ... We are investigating what may have happened," deputy police inspector Carsten Jansson from the Copenhagen police district of Vestegnen told AFP.
The book -- which Rasmussen says is a work of fiction -- reportedly features a celebrity magazine accessing private credit card information for stories about members of the royal family.
Niels Pinborg, who took over as editor of Se & Hoer in November last year, said that during his tenure "none of the things alleged in the book took place", but that owner Aller Media would investigate the claims.
"We view this very seriously and we have launched an internal investigation to find out if this could have happened," Nets spokesman Soeren Winge said.
These include coverage of Prince Joachim's 2008 honeymoon in Canada, according to news agency Ritzau.
The credit card transactions of former Danish Prime Minister Lars Loekke Rasmussen were also said to have been monitored.
Media mogul Rupert Murdoch shut down British tabloid News of the World in July 2011 following allegations of widespread phone hacking at the Sunday paper.
Following a major public inquiry into the British press in the wake of the scandal, a cross-party plan for press regulation was agreed in October, proposing new rules and fines for errant editors.
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