The 24-year-old sealed a £23.6 million move to Everton from Ajax in June, with BBC Sport reporting the figures involved.
And The Liverpool Echo has claimed that the Dutch ace received a two-word text message from the Arsenal icon, which read "Good choice".
Klaassen, who has four goals in 14 international appearances, has only started one of Everton's first three Premier League games so far.
Even in pre-season, the ex-Ajax man hasn't looked outstanding but it's a very different league compared to the Eredivisie and he will need time to adapt to England's top flight.
Klaassen worked with Bergkamp, now a coach, in the Dutch capital and the legendary forward will be fully aware about his capabilities.
Everton host Tottenham at Goodison Park on Saturday and Klaassen, if he is fit, will hope to be involved.