According to The Mirror, the Blues look set to see off competition from Premier League rivals Arsenal and Manchester City to acquire a player who rejected a move to Stamford Bridge during the summer transfer window in order to gain more first-team experience.
However, Arp's representative, Jurgen Milewski, has quickly dismissed those rumours. He told German publication Bild: "There is nothing in the story."
A product of Hamburg's academy, Arp became the first player born in the 21st century - January 2000 - to appear in the Bundesliga when he came off the bench in September to make a senior debut for his boyhood idols in their goalless derby draw with rivals Werder Bremen.
The 17-year-old forward made further history five weeks later by scoring on his next substitute appearance against Hertha Berlin, with the goal earning him a spot in head coach Markus Gisdol starting XI for subsequent meetings with Stuttgart, Schalke and Hoffenheim.
Arp, who first turned out for Hamburg at Under-11 level, scored 26 goals in 21 games in the youth setup last term and has also impressed for his country in 2017, notching seven - including the quickest hat-trick in tournament history - to help Germany reach the semi-finals of the European Under-17 Championship in May.
Last month, he netted five times in as many matches at the Under-17 World Cup as Christian Wuck's side bowed out at the quarter-final stage in India courtesy of a 2-1 defeat to Brazil.
Antonio Conte and Chelsea were initially said to be chasing Arp earlier this year, only for the player, also believed to be on the radar of several other European heavyweights including Real Madrid and Juventus, to rule out a premature move abroad while showing maturity beyond his years.
"It won't make any sense to leave my team and move abroad at the moment. I hardly achieved anything and have virtually no experience in professional football," he said in August. "The risk would have been too big. Money shouldn't matter at my age."
Arp, who has previously admitted to viewing England's Harry Kane as a role model while admiring the path he has taken to success at Tottenham Hotspur, is currently under contract until 2019 having penned a two-year extension over the summer.
Hamburg are now hoping to secure his long-term future while continuing to keep the mounting transfer interest at bay.