Clare Haughey's 20-year-old son Charlie was found dead on Saturday in the city. The cause of death is not yet known.
His older brother Sean paid tribute to Charlie in a public Facebook post, saying the family are "all devastated at the loss".
Police in Amsterdam have now confirmed they are investigating whether the death of a 20-year-old man who lived in Scotland was drug-related.
Ms Haughey was elected to represent Rutherglen in 2016 and was appointed Mental Health Minister in June 2018.
SNP group convener Bruce Crawford said: "This is devastating news and our thoughts are with our dear friend and colleague Clare, her family and all those who knew Charlie.
"The family is very grateful for the warm messages received and kindly request that their privacy is respected at this time."
A Foreign Office spokesman said: "We are supporting the family of a man following his death in Amsterdam and our staff are in contact with the police."
First Minister and SNP leader Nicola Sturgeon also expressed her condolences.
She tweeted: "My thoughts - and those of all of us in @theSNP - are with Clare and her family at this desperately sad time.
"I know they will be grateful for privacy as they try to come to terms with their loss."