Online footage sparked speculation that cocaine was on the acts table during the event.
David has volunteered to be tested on his return to Italy.
The Italian rock band won the 2021 competition with their song Zitti e Buoni, getting 524 points.
Speaking at a press conference, the singer said: “I don’t do drugs. Please, guys. Don’t say that really, no cocaine.”
Eurovision checked footage and table
The footage online saw Damanio David in the competition green room alongside guitarist Victoria De Angelis, musician Thomas Raggi and drummer Ethan Torchio.
The camera cut to the Italian delegation, and within the footage, 22-year-old David could be seen lowering his head behind a bucket of drinks, sparking claims that he was snorting cocaine.
In a statement, Eurovision said the singer would "take a voluntary drug test after arriving home".
Speaking on BBC Breakfast, Eurovision Song Contest Executive Supervisor Martin Osterdahl, said they have conducted tests on the bands table, and checked the footage.
“It’s very unfortunate, and we will have to see how this plays out in the end,” Osterdahl said.
“We have checked what we can check. We have checked the footage. We have checked out the glass, and we did find glass cells, so that supports the band’s story.
“The band was very upset about this, and they wanted to do a drug test immediately. We couldn’t provide that on the night in the area. So they said they will do it straight away when they get back.“
The rock band became the first group act to win the singing competition since 2006, and also gave Italy its first victory since 1990. The UK came last, with James Newman’s embers scoring nul points.
In a statement released by the Song Contest, it said: "The band, their management and head of delegation have informed us that no drugs were present in the Green Room and explained that a glass was broken at their table and it was being cleared by the singer. The EBU can confirm broken glass was found after an on site check.”
Maneskin reacted to the claims via Instagram, saying they were shocked and did not condone the use of drugs.
“We are really shocked about what some people are saying about Damiano doing drugs,” the band said.
"We really are AGAINST drugs and we never used cocaine. We are ready to get [tested], cause we have nothing to hide.
"We are here to play our music and we are so happy about our Eurovision win and we wanna thank everyone for supporting us. Rock'n Roll never dies. We love you."